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Problems and SolutionsinHilbert space theory,Fourier transform,waveletsandgeneralized functions

byWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South Africa

Preface

The purpose of this book is to supply a collection of problems in Hilbertspace theory, wavelets and generalized functions.

Prescribed books for problems.

1) Hilbert Spaces, Wavelets, Generalized Functions and Modern QuantumMechanics

by Willi-Hans SteebKluwer Academic Publishers, 1998ISBN 0-7923-5231-9

2) Classical and Quantum Computing with C++ and Java Simulations

by Yorick Hardy and Willi-Hans SteebBirkhauser Verlag, Boston, 2002ISBN 376-436-610-0

3) Problems and Solutions in Quantum Computing and Quantum Informa-tion, second edition

by Willi-Hans Steeb and Yorick HardyWorld Scientific, Singapore, 2006ISBN 981-256-916-2http://www.worldscibooks.com/physics/6077.html

The International School for Scientific Computing (ISSC) provides certifi-cate courses for this subject. Please contact the author if you want to dothis course or other courses of the ISSC.

e-mail addresses of the author:

[email protected][email protected]

Home page of the author:

http://issc.uj.ac.za

v

http://www.worldscibooks.com/physics/6077.html

Contents

Notation x

1 General 11.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 9

2 Finite Dimensional Hilbert Spaces 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 20

3 Hilbert Space L2(Ω) 223.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 49

4 Hilbert Space `2(N) 544.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 57

5 Fourier Transform 585.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 61

6 Wavelets 636.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 66

7 Linear Operators 687.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 687.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 71

vii

8 Generalized Functions 738.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 738.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 85

Bibliography 90

Index 93

viii

Notation

:= is defined as∈ belongs to (a set)/∈ does not belong to (a set)∩ intersection of sets∪ union of sets∅ empty setN set of natural numbersZ set of integersQ set of rational numbersR set of real numbersR+ set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidean space

space of column vectors with n real componentsCn n-dimensional complex linear space

space of column vectors with n complex componentsH Hilbert spacei

√−1

<z real part of the complex number z=z imaginary part of the complex number z|z| modulus of complex number z

|x+ iy| = (x2 + y2)1/2, x, y ∈ RT ⊂ S subset T of set SS ∩ T the intersection of the sets S and TS ∪ T the union of the sets S and Tf(S) image of set S under mapping ff g composition of two mappings (f g)(x) = f(g(x))x column vector in CnxT transpose of x (row vector)0 zero (column) vector‖ . ‖ normx · y ≡ x∗y scalar product (inner product) in Cnx× y vector product in R3

A,B,C m× n matricesdet(A) determinant of a square matrix Atr(A) trace of a square matrix Arank(A) rank of matrix AAT transpose of matrix A

x

A conjugate of matrix AA∗ conjugate transpose of matrix AA† conjugate transpose of matrix A

(notation used in physics)A−1 inverse of square matrix A (if it exists)In n× n unit matrixI unit operator0n n× n zero matrixAB matrix product of m× n matrix A

and n× p matrix BA •B Hadamard product (entry-wise product)

of m× n matrices A and B[A,B] := AB −BA commutator for square matrices A and B[A,B]+ := AB +BA anticommutator for square matrices A and BA⊗B Kronecker product of matrices A and BA⊕B Direct sum of matrices A and Bδjk Kronecker delta with δjk = 1 for j = k

and δjk = 0 for j 6= kδ delta functionΘ Heaviside’s functionλ eigenvalueε real parametert time variableH Hamilton operator

xi

Chapter 1

General

1.1 Solved ProblemsProblem 1. Consider a Hilbert space H with scalar product 〈 , 〉. Thescalar product implies a norm via ‖f‖2 := 〈f, f〉, where f ∈ H.(i) Show that

‖f + g‖2 + ‖f − g‖2 = 2(‖f‖2 + ‖g‖2).

Start with

‖f + g‖2 + ‖f − g‖2 = 〈f + g, f + g〉+ 〈f − g, f − g〉.

(ii) Assume that 〈f, g〉 = 0, where f, g ∈ H. Show that

‖f + g‖2 = ‖f‖2 + ‖g‖2.

Start with‖f + g‖2 = 〈f + g, f + g〉.

Problem 2. Show that

〈f, g〉 =14‖f + g‖2 − 1

4‖f − g‖2

or〈f, g〉 =

14‖f + g‖2 − 1

4‖f − g‖2 +

i

4‖f + ig‖2 − i

4‖f − ig‖2

1

2 Problems and Solutions

depending on whether we are dealing with a real or complex Hilbert space.

Problem 3. Let H be a Hilbert space with scalar product 〈 , 〉. Let u, v ∈H. Let ‖.‖ be the norm induced by the scalar product, i.e. ‖u‖2 = 〈u, u〉.Show that (Schwarz-Cauchy inequality))

|〈u, v〉| ≤ ‖u‖ · ‖v‖.

Obviously for u = 0 or v = 0 the inequality is an equality. So we canassume that u 6= 0 and v 6= 0 in the following.

Problem 4. (i) Let α ∈ C and u, v ∈ H. Find

〈u+ αv, u+ αv〉.

(ii) Let u, v ∈ H and v 6= 0. Show that

〈u, v〉〈v, u〉 ≤ 〈u, u〉〈v, v〉.

Problem 5. Let f, g ∈ H. Show that

〈f |f〉〈g|g〉 ≥ 14

(〈f |g〉+ 〈g|f〉)2.

Problem 6. Let f, g ∈ H. Use the Schwarz inequality

|〈f, g〉|2 ≤ 〈f, f〉〈g, g〉 = ‖f‖2 ‖g‖2

to prove the triangle inequality

‖f + g‖ ≤ ‖f‖+ ‖g‖.

Problem 7. Consider a Hilbert space H and ‖.‖ be the norm implied bythe scalar product. Let u,v ∈ H.(i) Show that

‖u− v‖+ ‖v‖ ≥ ‖u‖.

(ii) Show that〈u,v〉+ 〈v,u〉 ≤ 2‖u‖ · ‖v‖.

Problem 8. Let P be a nonzero projection operator in a Hilbert spaceH. Show that ‖P‖ = 1.

General 3

Problem 9. Let |ψ〉, |s〉, |φ〉 be normalized states in a Hilbert space H.Let U be a unitary operator, i.e. U−1 = U∗ in the Hilbert space H ⊗ Hsuch that

U(|ψ〉 ⊗ |s〉) = |ψ〉 ⊗ |ψ〉U(|φ〉 ⊗ |s〉) = |φ〉 ⊗ |φ〉.

Show that 〈φ|ψ〉 = 〈φ|ψ〉2. Find solutions to this equation.

Problem 10. A family, ψj j∈J of vectors in the Hilbert space, H,is called a frame if for any f ∈ H there exist two constants k1 > 0 and0 < k2 <∞, such that

k1‖f‖2 ≤∑j∈J〈ψj |f〉|2 ≤ k2‖f‖2.

Consider the Hilbert space H = R2 and the family of vectorsψ0 =

(11

), ψ1 =

(1−1

).

Show that we have a tight frame.

Problem 11. Let T : X → Y be a linear map between linear spaces(vector spaces) X, Y . The null space or kernel of the linear map T , denotedby ker(T ), is the subset of X defined by

ker(T ) := x ∈ X : Tx = 0 .

The range of T , denoted by ranT , is the subset of Y defined by

ran(T ) := y ∈ Y : there exists x ∈ X such that Tx = y .

Let P be a projection operator in a Hilbert space H. Show that ran(P ) isclosed and

H = ran(P )⊕ ker(P )

is the orthogonal direct sum of ran(P ) and ker(P ).

Problem 12. Let H be an arbitrary Hilbert space with scalar product〈 , 〉. Show that if ϕ is a bounded linear functional on the Hilbert space H,then there is a unique vector u ∈ H such that

ϕ(x) = 〈u, x〉 for all x ∈ H.

Problem 13. Let H be an arbitrary Hilbert space. A bounded linear op-erator A : H → H satisfies the Fredholm alternative if one of the following


two alternatives holds:

(i) either Ax = 0, A∗x = 0 have only the zero solution, and the linearequations Ax = y, A∗x = y have a unique solution x ∈ H for every y ∈ H;(ii) or Ax = 0, A∗x = 0 have nontrivial, finite-dimensional solution spacesof the same dimension, Ax = y has a (nonunique) solution if and only ify ⊥ u for every solution u of A∗u = 0, and A∗x = y has a (nonunique)solution if and only if y ⊥ u for every solution u of Au = 0.

Give an example of a bounded linear operator that satisfies the Fredholmalternative.

Problem 14. Let (M,d) be a complete metric space (for example aHilbert space) and let f : M →M be a mapping such that

d(f (m)(x), f (m)(y)) ≤ kd(x, y), ∀x, y ∈M

for some m ≥ 1, where 0 ≤ k < 1 is a constant. Show that the map f hasa unique fixed point in M .

Problem 15. Let H be a Hilbert space and let f : H → H be a monotonemapping such that for some constant β > 0

‖f(u)− f(v)‖ ≤ β‖u− v‖ ∀u, v ∈ H.

Show that for any w ∈ H, the equation u+ f(u) = w has a unique solutionu.

Problem 16. Consider a Hilbert space H with scalar product 〈, 〉. Let u,v, w be elements of the Hilbert space with ‖u‖ = ‖v‖ = ‖w‖ = 1. Showthat √

1− |〈u,v〉|2 ≤√

1− |〈u,w〉|2 +√

1− |〈w,v〉|2.

Problem 17. Given a Hilbert space H and a Hilbert subspace G of H.The Hilbert space projection theorem states that for every f ∈ H, thereexists a unique g ∈ G such that

(i) f − g ∈ G⊥

(ii) ‖f − g‖ = infh∈G‖f − h‖

where the space G⊥ is defined by

G⊥ := k ∈ H : 〈k|u〉 = 0 for all u ∈ G .

General 5

Show that if g is the minimizer of ‖f − h‖ over all h ∈ G, then it is truethat f − g ∈ G⊥.

Problem 18. Let φn n∈Z be an orthonormal basis in a Hilbert spaceH. Then any vector f ∈ H can be written as

f =∑n∈Z〈f, φn〉φn.

Now suppose that ψn n∈Z is also a basis for H, but it is not orthonormal.Show that if we can find a so-called dual basis χn n∈Z satisfying

〈ψn|χm〉 = δ(n−m)

then for any vector f ∈ H, we have

f =∑n∈Z〈f |χn〉ψn.

Here δ(n − m) denotes the Kronecker delta with δ(n − m) = 0 if n = mand 1 otherwise.

Problem 19. Let (X1, ‖·‖1) and (X2, ‖·‖2) be two normed spaces. Showthat the product vector spaces X = X1×X2 is also a normed vector spaceif we define

‖x‖ := max(‖x1‖1, ‖x2‖2)

with x = (x1, x2).

Problem 20. Let A be a linear bounded self-adjoint operator in a Hilbertspace H. Let u, v ∈ H and λ ∈ C. Consider the equation

Au− λu = v.

(i) Show that for λ nonreal (i.e. it has an imaginary part) v cannot vanishunless u vanishes.(ii) Show that for λ nonreal we have

‖(A− λI)−1v‖ ≤ 1|=(λ)|

‖v‖.

Problem 21. Let E be the exterior of the unit disc

z ∈ C : |z| > 1

and T the unit circle z ∈ C : |z| = 1 .


Let H2(E) be the Hardy space of square integrable functions on T, analyticin the region E. The inner product for f(z), g(z) ∈ H2(E) is defined by

〈f, g〉 =1

2π

∫ π

−πf(eiω)∗g(eiω)dω =

12πi

∮Tf∗(1/z∗)g(z)

dz

z.

Let f(z) = z2 and g(z) = z + 1. Find the scalar product 〈f, g〉.

Problem 22. Let O = u1.u2, . . . be an orthonormal set in a infinitedimensional Hilbert space. Show that if

x =∞∑j=1

cjuj

then

‖x‖2 =∞∑j=1

|cj |2.

Problem 23. Two Cauchy sequences xk and yk are said to beequivalent if for all ε > 0, there is a k(ε) such that for all j ≥ k(ε) we haved(xj , yj) < ε. One writes xk ∼ yk . Obviously, ∼ is an equivalencerelationship. Show that equivalent Cauchy sequences have the same limit.

Problem 24. Consider the sequence xk , k = 1, 2, . . . in R definedby xk = 1/k2 for all k = 1, 2, . . .. Show that this sequence is a Cauchysequence.

Problem 25. Let H be a Hilbert space and S be a sub Hilbert space.Show that any element u of H can be decomposed uniquely

u = v + w

where v is in S and w is in S⊥.

Problem 26. Let u, v1, v2 be elements of a Hilbert space. Show that

2‖u− v1‖2 + 2‖u− v2‖2 = ‖2(u− v1 + v2

2

)‖2 + ‖v1 − v2‖2.

Problem 27. Let P be the set of prime numbers. We define the set

S := (p, q) : p, q ∈ P p ≤ q .

General 7

Show thatd((p1, q1), (p2, q2)) := |p1q1 − p2q2|

defines a metric.

Problem 28. Consider the vector space of all continuous functions de-fined on [a, b]. We define a metric

d(f, g) := maxa≤x≤b

|f(x)− g(x)|.

Let a = π, b = π, f(x) = sin(x) and g(x) = cos(x). Find d(f, g).

Problem 29. The n×n matrices over R form a vector space. Show that

d(A,B) :=n∑j=1

n∑k=1

|ajk − bjk|

defines a metric.

Problem 30. Let n ≥ 1. Consider the continuous function

fn(t) =

0 0 ≤ t < 1/2− 1/n1/2 + n

2 (t− 1/2) 1/2− 1/n ≤ t ≤ 1/2 + 1/n1 1/2 + 1/n ≤ t ≤ 1

Show that the sequence fn(t) is not a Cauchy sequence for the uniformnorm, but with any of the Lp norms (1 ≤ p <∞) it is a Cauchy sequence.

Problem 31. The sequence space consists of the set of all (bounded orunbounded) sequences of complex

x = (χ1, χ2, . . .)

Thus we have a vector space. Can we define a metric in this vector spacewhich is implied by a norm?

Problem 32. A sequence fn (n ∈ N) of elements in a normed spaceE is called a Cauchy sequence if, for every ε > 0, there exists a number Mε,such that ‖fp − fq‖ < ε for p, q > Mε. Consider the Hilbert space R. Showthat

sn =n∑j=1

1(j − 1)!

, n ≥ 1

is a Cauchy sequence.


Problem 33. Two Cauchy sequences xk and yk are said to beequivalent if for all ε > 0, there is a k(ε) such that for all j ≥ k(ε) we haved(xj , yj) < ε. One writes xk ∼ yk . Obviously, ∼ is an equivalencerelationship. Show that equivalent Cauchy sequences have the same limit.

Problem 34. Consider the sequence xk , k = 1, 2, . . . in R definedby xk = 1/k2 for all k = 1, 2, . . .. Show that this sequence is a Cauchysequence.

Problem 35. The sequence space consists of the set of all (bounded orunbounded) sequences of complex numbers

x = (x1, x2, . . .)

Thus we have a vector space. Can we define a metric in this vector spacewhich is not implied by a norm?

Problem 36. Consider a complex Hilbert spaceH and |φ1〉, |φ2〉 ∈ H. Letc1, c2 ∈ C. An antilinear operator K in this Hilbert spaceH is characterizedby

K(c1|φ1〉+ c2|φ2〉) = c∗1K|φ1〉+ c∗2K|φ2〉.

A comb is an antilinear operator K with zero expectation value for all states|ψ〉 of a certain complex Hilbert space H. This means

〈ψ|K|ψ〉 = 〈ψ|LC|ψ〉 = 〈ψ|L|ψ∗〉 = 0

for all states |ψ〉 ∈ H, where L is a linear operator and C is the complexconjugation.(i) Consider the two-dimensional Hilbert space H = C2. Find a unitary2× 2 matrix such that

〈ψ|UC|ψ〉 = 0.

(ii) Consider the Pauli spin matrices σ1, σ2, σ3 with σ0 = I2. Find

3∑µ=0

3∑ν=0

〈ψ|σµC|ψ〉gµ,ν〈ψ|σνC|ψ〉

where gµ,ν = diag(−1, 1, 0, 1).

General 9

1.2 Supplementary Problems

Problem 1. (i) Let f, g ∈ H. Find all solutions to the equations

〈f, g〉〈g, f〉 = 1.

(i) Let f, g ∈ H. Find all solutions to the equations

〈f, g〉〈g, f〉 = i.

Problem 2. Let H be a Hilbert space and f ∈ H. Let ek : k ∈ I(I index set) be an orthonormal sequence in the Hilbert space H. Then(Bessel inequality) ∑

k∈I

|〈f, ek〉| ≤ ‖f‖.

Consider the Hilbert space R3 and

f =1√3

111

, e1 =

101

, e2 =

10−1

.

Find the left-hand side and right-hand side of the inequality.

Problem 3. Let x, y ∈ R. Is

d(x, y) :=|x− y|

2 + |x− y|

a metric on R?

Problem 4. Show that each Cauchy sequence is bounded.

Chapter 2

Finite DimensionalHilbert Spaces

2.1 Introduction

We consider the Hilbert space Cn with the scalar product

v∗v =n∑j=1

vjvj

and the Hilbert space for n× n matrices over C with the scalar product

〈A,B〉 = tr(AB∗).

The standard basis in Cn is given by100...0

,

010...0

, · · ·

000...1

.

10

Finite Dimensional Hilbert Spaces 11

2.2 Solved Problems

Problem 1. Consider the Hilbert space R4 and the vectors1000

,

1100

,

1110

,

1111

.

(i) Show that the vectors are linearly independent.(ii) Use the Gram-Schmidt orthogonalization process to find mutually or-thogonal vectors.

Problem 2. Consider the Hilbert space R4. Show that the vectors (Bellbasis)

1√2

1001

,1√2

100−1

,1√2

0110

,1√2

01−10

are linearly independent. Show that they form a orthonormal basis in theHilbert space R4.

Problem 3. Consider the Hilbert space R4. Find all pairwise orthogonalvectors (column vectors) x1, . . . ,xp, where the entries of the column vectorscan only be +1 or −1. Calculate the matrix

p∑i=1

xixTi

and find the eigenvalues and eigenvectors of this matrix

Problem 4. Consider the Hilbert space C2 and the vectors

|0〉 =(ii

), |1〉 =

(1−1

).

Normalize these vectors and then calculate the probability |〈0|1〉|2.

Problem 5. Consider the Hilbert space Rn. Let x,y ∈ Rn. Show that

‖x + y‖2 + ‖x− y‖2 ≡ 2(‖x‖2 + ‖y‖2).

Note that‖x‖2 := 〈x,x〉.


Problem 6. Let |0〉, |1〉 be an orthonormal basis in the Hilbert space C2.Let

|ψ〉 = cos(θ/2)|0〉+ eiφ sin(θ/2)|1〉

where θ, φ ∈ R.(i) Find 〈ψ|ψ〉.(ii) Find the probability |〈0|ψ〉|2. Discuss |〈0|ψ〉|2 as a function of θ.(iii) Assume that

|0〉 =(

10

), |1〉 =

(01

).

Find the 2× 2 matrix |ψ〉〈ψ| and calculate the eigenvalues.

Problem 7. Consider the Hilbert space R2. Show that the vectors1√2

(11

),

1√2

(1−1

)are linearly independent. Find

1√2

(11

)⊗ 1√

2

(11

),

1√2

(11

)⊗ 1√

2

(1−1

),

1√2

(1−1

)⊗ 1√

2

(11

),

1√2

(1−1

)⊗ 1√

2

(1−1

).

Show that these four vectors form a basis in R4. Consider the 4× 4 matrixQ which is constructed from the four vectors given above, i.e. the columnsof the 4 × 4 matrix are the four vectors. Find QT . Is Q invertible? If sofind the inverse Q−1. What is the use of the matrix Q?

Problem 8. Consider the Hilbert space R4. Let A be a symmetric 4× 4matrix over R. Assume that the eigenvalues are given by λ1 = 0, λ2 = 1,λ3 = 2 and λ4 = 3 with the corresponding normalized eigenfunctions

u1 =1√2

1001

, u2 =1√2

100−1

, u3 =1√2

0110

, u4 =1√2

01−10

.

Find the matrix A by means of the spectral theorem.

Problem 9. Show that the 2× 2 matrices

A =1√2

(1 00 1

), B =

1√2

(0 11 0

),


C =1√2

(0 −ii 0

), D =

1√2

(1 00 −1

)form an orthonormal basis in the Hilbert space M2(C).

Problem 10. Consider the Hilbert space H of the 2 × 2 matrices overthe complex numbers with the scalar product

〈A,B〉 := tr(AB∗), A,B ∈ H.

Show that the rescaled Pauli matrices µj := 1√2σj , j = 1, 2, 3

µ1 =1√2

(0 11 0

), µ2 =

1√2

(0 −ii 0

), µ3 =

1√2

(1 00 −1

)plus the rescaled 2× 2 identity matrix

µ0 =1√2

(1 00 1

)form an orthonormal basis in the Hilbert space H.

Problem 11. Let A, B be two n× n matrices over C. We introduce thescalar product

〈A,B〉 :=tr(AB∗)

trIn=

1n

tr(AB∗).

This provides us with a Hilbert space.The Lie group SU(N) is defined by the complex n× n matrices U

SU(N) := U : U∗U = UU∗ = In , det(U) = 1 .

The dimension is N2 − 1. The Lie algebra su(N) is defined by the n × nmatrices X

su(N) := X : X∗ = −X , tr(X) = 0 .

(i) Let U ∈ SU(N). Calculate 〈U,U〉.(ii) Let A be an arbitrary complex n×n matrix. Let U ∈ SU(N). Calculate〈UA,UA〉.(iii) Consider the Lie algebra su(2). Provide a basis. The elements of thebasis should be orthogonal to each other with respect to the scalar productgiven above. Calculate the commutators of these matrices.

Problem 12. Let H = ωS1 be a Hamilton operator, where

S1 :=~√2

0 1 01 0 10 1 0


and ω is the frequency.(i) Find exp(−iHt/~)ψ(0), where ψ(0) = (1, 1, 1)T /

√3.

(ii) Calculate the time evolution of

S3 := ~

1 0 00 0 00 0 −1

using the Heisenberg equation of motion. The matrices Sx, Sy, Sz are thespin-1 matrices, where

S2 :=~√2

0 −i 0i 0 −i0 i 0

.

Problem 13. Consider the Hilbert space R4. Show that the Bell basis

u1 =1√2

1001

, u2 =1√2

100−1

, u3 =1√2

0110

, u4 =1√2

01−10

forms an orthonormal basis in this Hilbert space.

Problem 14. (i) Consider the Hilbert space C4. Show that the matrices

Π1 =12

(I2 ⊗ I2 + σ1 ⊗ σ1), Π2 =12

(I2 ⊗ I2 − σ1 ⊗ σ1)

are projection matrices in C4.(ii) Find Π1Π2. Discuss.(iii) Let e1, e2, e3, e4 be the standard basis in C4. Calculate

Π1ej , Π2ej , j = 1, 2, 3, 4

and show that we obtain 2 two-dimensional Hilbert spaces under theseprojections.

Problem 15. Consider the 3× 3 matrix

A =

2 0 21 0 00 0 1

.

(i) The matrix A can be considered as an element of the Hilbert space ofthe 3 × 3 matrices with the scalar product 〈A,B〉 := tr(ABT ). Find thenorm of A with respect to this Hilbert space.


(ii) On the other hand A can be considered as a linear operator in theHilbert space R3. Find die norm

‖A‖ := sup‖x‖=1

‖Ax‖, x ∈ R3.

(iii) Find the eigenvalues of A and AAT . Compare the result with (i) and(ii).

Problem 16. Consider the Hilbert space M4(C) of all 4×4 matrices overC with the scalar product 〈A,B〉 := tr(AB∗), where A,B ∈ M4(C). Theγ-matrices are given by

γ1 =

0 0 0 −i0 0 −i 00 i 0 0i 0 0 0

, γ2 =

0 0 0 −10 0 1 00 1 0 0−1 0 0 0

γ3 =

0 0 −i 00 0 0 ii 0 0 00 −i 0 0

, γ4 =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

and

γ5 = γ1γ2γ3γ4 =

0 0 −1 00 0 0 −1−1 0 0 00 −1 0 0

.

We define the 4× 4 matrices

σjk :=i

2[γj , γk], j < k

where j = 1, 2, 3, k = 2, 3, 4 and [ , ] denotes the commutator.(i) Calculate σ12, σ13, σ14, σ23, σ24, σ34.(ii) Do the 16 matrices

I4, γ1, γ2, γ3, γ4, γ5, γ5γ1, γ5γ2, γ5γ3, γ5γ4, σ12, σ13, σ14, σ23, σ24, σ34

form a basis in the Hilbert space M4(C)? If so is the basis orthogonal?

Problem 17. Find the spectrum (eigenvalues and normalized eigenvec-tors) of matrix

A =

1 1 11 1 11 1 1

.


Find ‖A‖, where ‖.‖ denotes the norm.

Problem 18. Let A and B be two arbitrary matrices. Give the definitionof the Kronecker product. Let uj (j = 1, 2, . . . ,m) be an orthonormal basisin the Hilbert space Rm. Let vk (k = 1, 2, . . . , n) be an orthonormal basis inthe Hilbert space Rn. Show that uj⊗vk (j = 1, 2, . . . ,m), (k = 1, 2, . . . , n)is an orthonormal basis in Rm+n.


A =1√2

(1 00 1

), B =

1√2

(0 11 0

),

C =1√2

(0 −ii 0

), D =

1√2

(1 00 −1

)form an orthonormal basis in the Hilbert space M2(C).


A =(

1 00 0

), B =

(1 10 0

), C =

(1 11 0

), D =

(1 11 1

)form a basis in the Hilbert space M2(R). Apply the Gram-Schmidt tech-nique to obtain an orthonormal basis.

Problem 21. Consider the 3× 3 matrices over the real numbers

A =

2 0 21 0 00 0 1

.

(i) The matrix A can be considered as an element of the Hilbert space ofthe 3× 3 matrices over the real nunbers with the scalar product

〈B,C〉 := tr(BCT ).

Find the norm of A with respect to this Hilbert space.(ii) On the other hand the matrix A can be considered as a linear operatorin the Hilbert space R3. Find the norm

‖A‖ := sup‖x‖=1

‖Ax‖, x ∈ R3.

(iii) Find the eigenvalues of A and ATA. Compare the result with (i) and(ii).


Problem 22. Consider the Hilbert space C2. The Pauli spin matricesσx, σy, σz act as linear operators in this Hilbert space. Let

H = ~ωσ3

be a Hamilton operator, where

σ1 =(

1 00 −1

)and ω is the frequency. Calculate the time evolution (intial value problem)of

σ1 =(

0 11 0

)i.e.

i~dσ1

dt= [σ1, H](t).

Problem 23. Consider the Hilbert space C4. Consider the Hamiltonoperator

H := ~ω

0 0 0 −i0 0 −i 00 i 0 0i 0 0 0

.

Find the time-evolution of the operator

γ3 =

0 0 −i 00 0 0 ii 0 0 00 −i 0 0

using the Heisenberg equation of motion

i~dγ3

dt= [γ3, H](t).

Problem 24. Let M be any n × n matrix. Let x = (x1, x2, . . .)T . Thelinear operator A is defined by

Ax = (w1, w2, . . .)T

where

wj :=n∑k=1

Mjkxk, j = 1, 2, . . . , n

wj := xj , j > n


and D(A) = `2(N). Show that A is self-adjoint if the n × n matrix M ishermitian. Show that A is unitary if M is unitary.

Problem 25. Consider the Hilbert space Cn. Let uj , j = 1, 2, . . . , n, andvj , j = 1, 2, . . . , n be orthonormal bases in Cn, where uj , vj are consideredas column vectors. Show that

U =n∑j=1

ujv∗j

is a unitary n× n matrix.

Problem 26. Consider the Hilbert space R2. Given the vectors

u1 =(

01

), u2 =

(√3/2−1/2

), u3 =

(−√

3/2−1/2

).

The three vectors u1, u2, u3 are at 120 degrees of each other and arenormalized, i.e. ‖uj‖ = 1 for j = 1, 2, 3. Every given two-dimensionalvector v can be written as

v = c1u1 + c2u2 + c3u3, c1, c2, c3 ∈ R

in many different ways. Given the vector v minimize

12

(c21 + c22 + c23)

subject to the constraint v − c1u1 − c2u2 − c3u3 = 0.

Problem 27. Let A, H be n× n hermitian matrices, where H plays therole of the Hamilton operator. The Heisenberg equations of motion is givenby

dA(t)dt

=i

~[H,A(t)].

with A = A(t = 0) = A(0). Let Ej (j = 1, 2, . . . , n2) be an orthonormalbasis in the Hilbert space H of the n× n matrices with scalar product

〈X,Y 〉 := tr(XY ∗), X, Y ∈ H.

Now A(t) can be expanded using this orthonormal basis as

A(t) =n2∑j=1

cj(t)Ej


and H can be expanded as

H =n2∑j=1

hjEj .

Find the time evolution for the coefficients cj(t), i.e. dcj/dt, where j =1, 2, . . . , n2.

Problem 28. Consider the Hilbert space M2(C) of all 2×2 matrices overC with scalar product

〈A,B〉 := tr(AB∗), A,B ∈M2(C).

The standard basis is

E11 =(

1 00 0

), E12 =

(0 10 0

), E21 =

(0 01 0

), E22 =

(0 00 1

).

A mutually unbiased basis is

µ0 =1√2σ0 =

1√2

(1 00 1

), µ1 =

1√2σ1 =

1√2

(0 11 0

),

µ2 =1√2σ2 =

1√2

(0 −ii 0

), µ3 =

1√2σ3 =

1√2

(1 00 −1

).

(i) Express the Hadamard matrix

A =1√2

(1 11 −1

)with this mutually unbiased basis.(ii) Express the Bell matrix

B =1√2

1 0 0 10 1 1 00 1 −1 01 0 0 −1

with the basis (sixteen dimensional) given by

µj ⊗ µk, j, k = 0, 1, 2, 3.



Problem 1. Consider the eigenvalue problem

Av = λv.

Show thatλ =

v∗Avv∗v

.

Problem 2. Consider the hilbert space R4. Let x ∈ R. Do the vectors1xx2

x3

,

01

2x3x2

,

002

6x

.

0006

form a basis in this Hilbert space?

Problem 3. Consider the linear operator

A =

2 0 00 0 10 1 0

in the Hilbert space R3. Find

‖A‖ := sup‖x‖=1

‖Ax‖

using the method of the Lagrange multiplier.

Problem 4. Consider the Hilbert space R3. Let x ∈ R3, where x isconsidered as a column vector and x 6= 0. Find the matrix xxT and thereal number xTx. Show that the matrix xxT admits only one nonzeroeigenvalue given by xTx.

Problem 5. Consider the Hilbert space R3. Find the spectrum (eigen-values and normalized eigenvectors) of matrix

A =

1 2 31 2 31 2 3

.

Find ‖A‖ := supx=1 ‖Ax‖, where ‖.‖ denotes the norm and x ∈ R3.


Problem 6. Find the spectrum (eigenvalues and normalized eigenvectors)of the 3× 3 matrix

A =

3 3 33 3 33 3 3

.

Find ‖A‖, where ‖.‖ denotes the norms

‖A‖1 := sup‖x‖=1

‖Ax‖

‖A‖2 :=√

tr(AA∗).

Compare the norms with the eigenvalues. Find exp(A).

Problem 7. Letv0, v1, v2, v3

be an orthonormal basis in the Hilbert space C4. Show that the vectors

u0 =12

(v0 + v1 + v2 + v3), u1 =12

(v0 − v1 + v2 − v3),

u2 =12

(v0 + v1 − v2 − v3), u3 =12

(v0 − v1 − v2 + v3),

also form an orthonormal basis in C4.

Chapter 3

Hilbert Space L2(Ω)

Let Ω be a convex domain in Rn. We consider the Hilbert space of thesquare integrable function with scalar product

〈f, g〉 =∫

Ω

f(x)g∗(x)dx.

The scalar product implies the norm

‖f‖2 = 〈f, f〉 =∫

Ω

f(x)f∗(x)dx ≥ 0.

Consider the vector space

H(Cn×N ) :=f : Cn×N → C | f holomorphic

∫Cn×N

|f(z)|2dµ(z) <∞

where z = (zjk) with zjk = xjk + iyjk (j = 1, . . . , n; k = 1, . . . , N) and

dµ(z) =1

πnNexp(−tr(zz∗)), dz =

n∏j=1

N∏k=1

dxjkdyjk.

Then H(Cn×N ) is a Hilbert space with respect to the inner product

〈f, g〉 =∫

Cn×N

f(z)g(z)dµ(z).

22

Hilbert Space L2(Ω) 23

3.1 Solved ProblemsProblem 1. Consider the Hilbert space L2[a, b], where a, b ∈ R and b > a.Find the condition on a and b such that

〈cos(x), sin(x)〉 = 0

where 〈 , 〉 denotes the scalar product in L2[a, b].Hint. Since b > a, we can write b = x+ ε, where ε > 0.

Problem 2. Consider the Hilbert space L2[−1, 1]. Normalize the functionf(x) = x in this Hilbert space.

Problem 3. Consider the Hilbert space L2[−π, π]. Obviously cos(x) ∈L2[−π, π]. Find the norm ‖ cos(x)‖. Find nontrivial functions f, g ∈L2[−π, π] such that

〈f(x), cos(x)〉 = 0, 〈g(x), cos(x)〉 = 0

and〈f(x), g(x)〉 = 0.

Problem 4. Consider the Hilbert space L2[0, π]. Let ‖ ‖ be the norminduced by the scalar product of L2[0, π]. Find the constants a, b such that

‖ sin(x)− (ax2 + bx)‖

is a minimum.

Problem 5. Consider the Hilbert space L2[0, 1]. Find a non-trivial func-tion

f(x) = ax3 + bx2 + cx+ d.

such that

〈f(x), x〉 = 0, 〈f(x), x2〉 = 0, 〈f(x), x3〉 = 0

where 〈 , 〉 denotes the scalar product

Problem 6. (i) Consider the functions

f(x) =1

1 + x2, g(x) =

x

1 + x2.

Obviously f, g ∈ L2(R). Calculate the scalar product

〈f, g〉 =∫ ∞−∞

f(x)g(x)dx.


(ii) Let ω > 0. Consider the functions

f(t) =sin(ωt)ωt

, g(t) =1− cos(ωt)

ωt.

Obvioulsy f(0) = 1, g(0) = 0 and f, g ∈ L2(R). Calculate the scalarproduct

〈f, g〉 =∫ ∞−∞

f(t)g(t)dt.

Problem 7. Consider the Hilbert space L2(R). Let j, k = 1, 2, . . .. Con-sider the functions

fj(x) = xje−|x|, fk(x) = xke−|x|.

Find the scalar product

(fj(x), fk(x)) :=∫ ∞−∞

fj(x)fk(x)dx =∫ ∞−∞

fj(x)fk(x)dx.

Note that (a > 0 and n ∈ N)∫ ∞0

xne−axdx =Γ(n+ 1)an+1

.

Discuss.

Problem 8. A basis in the Hilbert space L2[0, 1] is given by

B :=e2πixn : n ∈ Z

.

Let

f(x) =

2x 0 ≤ x < 1/22(1− x) 1/2 ≤ x < 1

Is f ∈ L2[0, 1]? Find the first two expansion coefficients of the Fourierexpansion of f with respect to the basis given above.

Problem 9. (i) Consider the Hilbert space L2[−1, 1]. Consider the se-quence

fn(x) =

−1 if −1 ≤ x ≤ −1/nnx if −1/n ≤ x ≤ 1/n+1 if 1/n ≤ x ≤ 1

where n = 1, 2, . . .. Show that fn(x) is a sequence in L2[−1, 1] that is aCauchy sequence in the norm of L2[−1, 1].


(ii) Show that fn(x) converges in the norm of L2[−1, 1] to

sgn(x) =−1 if −1 ≤ x < 0+1 if 0 < x ≤ 1 .

(iii) Use this sequence to show that the space C[−1, 1] is a subspace ofL2[−1, 1] that is not closed.

Problem 10. Consider the Hilbert space L2[0, 1]. The Legendre polyno-mials are defined as

P0(x) = 1, Pn(x) =1

2nn!dn

dxn(x2 − 1).

Show that the first first four elements are given by

P0(x) = 1, P1(x) = x, P2(x) =12

(3x2 − 1), P3(x) =12

(5x3 − 3x).

Normalize the four elements. Show that the four elements are pairwiseorthonormal.

Problem 11. Let R be a bounded region in n-dimensional space. Con-sider the eigenvalue problem

−∆u = λu, u(q ∈ ∂R) = 0

where ∂R denotes the boundary of R.(i) Show that all eigenvalues are real and positive(ii) Show that the eigenfunctions which belong to different eigenvalues areorthogonal.

Problem 12. Consider the inner product space

C[a, b] = f(x) : f is continuous on x ∈ [a, b]

with the inner product

〈f, g〉 :=∫ b

a

f(x)g∗(x)dx.

This implies a norm

〈f, f〉 =∫ b

a

f(x)f∗(x)dx = ‖f‖2.

Show that C[a, b] is incomplete. This means find a Cauchy sequence inthe space C[a, b] which converges to an element which is not in the spaceC[a, b].


Problem 13. Consider the Hilbert space L2[−π, π]. Given the function

f(x) =

1 0 < x ≤ π0 x = 0−1 −π ≤< 0

Obviously f ∈ L2[−π, π]. Find the Fourier expansion of f . The orthonor-mal basis B is given by

B :=φk(x) =

1√2π

exp(ikx) k ∈ Z.

Find the approximation a0φ0(x) + a1φ1(x) + a−1φ−1(x), where a0, a1, a−1

are the Fourier coefficients.

Problem 14. Consider the linear operator A in the Hilbert space L2[0, 1]defined by Af(x) := xf(x). Find the matrix elements

〈Pi, APj〉

for i, j = 0, 1, 2, 3, where Pi are the (normalized) Legrende polynomials. Isthe matrix Aij symmetric?

Problem 15. Consider the Hilbert space L2[0, 2π). Let

g(x) = cos(x), f(x) = x.

Find the conditions on the coefficients of the polynomial

p(x) = a3x3 + a2x

2 + a1x+ a0

such that〈g(x), p(x)〉 = 0, 〈f(x), p(x)〉 = 0.

Solve the equations for a3, a2, a1, a0.

Problem 16. Let b > a. Consider the Hilbert space L2([a, b]) and thefunctions

φn(x) := sin(nπ

x− ab− a

), n = 1, 2, . . .

which form an orthonormal basis in L2([a, b]). Find

〈φm(x), xφn(x)〉 ≡∫ b

a

φm(x)xφn(x)dx, m, n = 0, 1, 2, . . .


Problem 17. Let m,n ∈ N. Consider the Hilbert space L2([−1, 1]) thefunctions

fn(x) =1

1 + nx2, n = 1, 2, . . .

which are elements in this Hilbert space. Find

‖fn(x)− fm(x)‖.

Problem 18. Consider the Hilbert space L2(R). Give the definitionand an example of an even function in L2(R). Give the definition and anexample of an odd function in L2(R). Show that any function f ∈ L2(R)can be written as a combination of an even and an odd function.

Problem 19. The Chebyshev polynomials Tn(x) of the 1-st kind aredefined for x ∈ [−1, 1] and given by

Tn(x) = cos(n arccosx), n = 0, 1, 2, . . .

The Chebyshev polynomials Un(x) of the 2-nd kind are defined for x ∈[−1, 1] and given by

Un(x) =sin((n+ 1) arccosx)√

1− x2, n = 0, 1, 2, . . . .

Consider the Hilbert spaces

H1 = L2

([−1, 1],

dx

π√

1− x2

), H2 = L2

([−1, 1],

2√

1− x2dx

π

)

which bases are formed by the Chebyshev polynomials of the 1-st and 2-ndtype

Φ(1)n (x) =

√2Tn(x), n ≥ 1, Φ(1)

0 = T0(x) = 1

Φ(2)n (x) =Un(x), n ≥ 0

Find a recursion relation for Φ(1)n and Φ(2)

n .

Problem 20. Consider the Hilbert space L2[0, 1]. Find a non-trivialpolynomial p

p(x) = ax3 + bx2 + cx+ d

such that〈p, 1〉 = 0, 〈p, x〉 = 0, 〈p, x2〉 = 0.


Problem 21. Consider the set of polynomials

1, x, x2, . . . , xn, . . . .

Use the Gram-Schmidt procedure and the inner product

〈f, g〉 =∫ b

a

f(x)g(x)ω(x)dx, ω(x) > 0

to obtain the first four orthogonal polynomials when(i) a = −1, b = 1, ω(x) = 1 (Legrendre polynomials)(ii) a = −1, b = 1, ω(x) = (1− x2)−1/2 (Chebyshev polynomials)(iii) a = 0, b = +∞, ω(x) = e−x (Laguerre polynomials)(iv) a = −∞, b = +∞, ω(x) = e−x

2(Hermite polynomials)

Problem 22. Consider the function

f(x) =∞∑j=0

12j

cos(jx).

Is f an element of L2[−π, π]?

Problem 23. Consider the Hilbert space L2([0, 1]). The shifted Legrendrepolynomials , defined on the interval [0, 1], are obtained from the Legren-dre polynomial by the transformation y = 2x − 1. The shifted Legrendrepolynomials are given by the recurrence formula

Pj(x) =(2j + 1)(2x− 1)

j + 1Pj(x)− j

j + 1Pj−1(x) j = 1, 2, . . .

and P0(x) = 1, P1(x) = 2x − 1. They are elements of the Hilbert spaceL2([0, 1]). A function u in the Hilbert space L2([0, 1]) can be approximatedin the form of a series with n+ 1 terms

u(x) =n∑j=0

cjPj(x)

where the coeffients cj ∈ R, j = 0, 1, . . . , n. Consider the Volterra integralequation of first kind

λ

∫ x

0

y(t)(x− t)α

dt = f(x), 0 ≤ t ≤ x ≤ 1

with 0 < α < 1 and f ∈ L2([0, 1]). Consider the ansatz

yn(x) = a0xα +

n∑j=0

cjPj(x).


to find an approximate solution to the Volterra integral equation of firstkind (α = 1/2)

λ

∫ x

0

y(t)√x− t

dt = f(x)

where

f(x) =2

105√x(105− 56x2 + 48x3).

Problem 24. Consider the Hilbert space L2(R). Let

fn(x) =x

1 + nx2, n = 1, 2, . . . .

(i) Find ‖fn(x)‖ andlimn→∞

‖fn(x)‖.

(ii) Does the sequence fn(x) converge uniformely on the real line?

Problem 25. Let n = 1, 2, . . .. We define the functions fn ∈ L2[0,∞) by

fn(x) =√

n for n ≤ x ≤ n+ 1/n0 otherwise

(i) Calculate the norm ‖fn − fm‖ implied by the scalar product. Does thesequence fn converge in the L2[0,∞) norm?(ii) Show that fn(x) converges pointwise in the domain [0,∞) and find thelimit. Does the sequence converge pointwise uniformly?(iii) Show that fn (n = 1, 2, . . .) is an orthonormal system. Is it a basisin the Hilbert space L2[0,∞)?

Problem 26. Consider the function f ∈ L2[0, 1]

f(x) =

x for 0 ≤ x ≤ 1/21− x for 1/2 ≤ x ≤ 1

A basis in the Hilbert space is given by

B :=

1,√

2 cos(πnx) : n = 1, 2, . . ..

Find the Fourier expansion of f with respect to this basis. From thisexpansion show that

π2

8=∞∑k=0

1(2k + 1)2

.


Problem 27. A particle is enclosed in a rectangular box with impene-trable walls, inside which it can move freely. The Hilbert space is

L2([0, a]× [0, b]× [0, c])

where a, b, c > 0. Find the eigenfunctions and the eigenvalues. What canbe said about the degeneracy, if any, of the eigenfunctions?

Problem 28. Consider the Hilbert space L2[0, 1] and the polynomials

1, x, x2, x3, x4 .

Apply the Gram-Schmidt orthogonalization process to these polynomials.

Problem 29. Consider the Hilbert space L2(T). Let f ∈ L2(T). Give anexample of a bounded linear functional.

Problem 30. Consider the Hilbert space L2(R). Show that the Hilbertspace is the direct sum of the Hilbert space M of even functions and theHilbert space N of odd functions. Give an example of such functions inthis Hilbert space.

Problem 31. Let a > 0. Consider the Hilbert space L2[0, a]. Let

Af(x) := xf(x)

for f ∈ L2[0, a]. Find the norm of the operator A. We define

‖A‖ := sup‖f‖=1

‖Af‖.

Problem 32. Consider the Hilbert space L2[0, 2π]. Let

g(x) = cos(x), f(x) = x.

Find the conditions on the coefficients of the polynomial

p(x) = a3x3 + a2x

2 + a1x+ a0

such that〈g(x), p(x)〉 = 0, 〈f(x), p(x)〉 = 0.

Solve the equations for a3, a2, a1, a0.

Problem 33. Consider the function f : R→ R

f(x) =1− cos(2πx)

x.


Using L’Hospital rules we have f(0) = 0. Is f ∈ L2(R)?

Problem 34. Consider the Hilbert space L2[−1, 1]. The Legendre poly-nomials are given by

Pj(x) :=1

2jj!dj

dxj(x2 − 1)j .

Find the scalar product〈Pj(x), Pk(x)〉.

Problem 35. Consider the Hilbert space H = L2(T). This is the vectorspace of 2π-periodic functions. Then

u(x) =1√2

is a constant function which is normalized, i.e. ‖u‖ = 1. Show that theprojection operator Pu defined by

Puf := 〈u, f〉u

maps a function f to its mean. This means

Puf = 〈f〉, 〈f〉 =∫ 2π

0

f(x)dx.

Problem 36. Consider the Hilbert space L2[−π, π] and the vector spaceof continiuous real-valued functions C[−π, π] on the interval [−π, π]. Letk > 0 and

fk(x) =

0 if −π ≤ x ≤ 0kx if 0 ≤ x ≤ 1/k1 if 1/k ≤ x ≤ π

The sequence of functions fk belongs to the vector space C[−π, π].(i) Show that fn → χ in the norm of the Hilbert space L2[−π, π], where

χ(x) :=

0 if−π ≤ x ≤ 01 if 0 < x ≤ π

so that the sequence fk is a Cauchy sequence in the Hilbert spaceL2[−π, π].(ii) Show that ‖χ− g‖ > 0 for every g ∈ C[−π, π]. Conclude that C[−π, π]is not a Hilbert space.


Problem 37. The Legrendre polynomials are defined on the interval[−1, 1] and defined by the recurrence formula

Lj(x) =2j + 1j + 1

xLj(x)− j

j + 1Lj−1(x) j = 1, 2, . . .

and L0(y) = 1, L1(x) = x. They are elements of the Hilbert spaceL2([−1, 1]). Calculate the scalar product

〈Lj(x), Lk(x)〉

for j, k = 0, 1, . . .. Discuss.

Problem 38. Let fn : [−1, 1]→ [−1, 1] be defined by

fn(x) =

1 for −1 ≤ x ≤ 0√

1− nx for 0 ≤ x ≤ 1/n0 for 1/n ≤ x ≤ 1

Show that fn ∈ L2[−1, 1]. Show that fn is a Cauchy sequence.

Problem 39. Consider the Hilbert space L2([−1, 1]). The Chebyshevpolynomials are defined by

Tn(x) := cos(n cos−1 x), n = 0, 1, 2, . . . .

They are elements of the Hilbert space L2([−1, 1]). We have

T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x.

Calculate the scalar products

〈T0, T1〉, 〈T1, T2〉, 〈T2, T3〉.

Calculate the integrals ∫ 1

−1

Tm(x)Tn(x)√1− x2

dx

for (m,n) = (0, 1), (m,n) = (1, 2), (m,n) = (2, 3).

Problem 40. (i) Consider the Hilbert space L2[0, 1] with the scalarproduct 〈·, ·〉. Let f : [0, 1]→ [0, 1]

f(x) :=

2x if x ∈ [0, 1/2)2(1− x) if x ∈ [1/2, 1]


Thus f ∈ L2[0, 1]. Calculate the moments µk, k = 0, 1, 2, . . . defined by

µk := 〈f(x), xk〉 ≡∫ 1

0

f(x)xkdx.

(ii) Show that∞∑k=0

|µk|2 < π

∫ 1

0

|f(x)|2dx.

Problem 41. Let a, b ∈ R and −∞ < a < b < +∞. Let f be a functionin the class C1 (i.e., the derivative df/dt exists and is continuous) on theinterval [a, b]. Thus f is also an element of the Hilbert space L2([a, b]).Show that

limω→∞

∫ b

a

f(t) sin(ωt)dt = 0. (1)

Problem 42. Consider the problem of a particle in a one-dimensionalbox. The underlying Hilbert space is L2(−a, a). Solve the Schrodingerequation

i~∂ψ

∂t= Hψ

as follows: The formal solution is given by

ψ(t) = exp(−iHt/~)ψ(0).

Expand ψ(0) with respect to the eigenfunctions of the operator H. Theeigenfunctions form a basis of the Hilbert space. Then apply exp(−iHt/~).Calculate the probability

P = |〈φ, ψ(t)〉|2

whereφ(q) =

1√a

sin(πqa

)and

ψ(q, 0) =1√a

sin(πqa

).

Problem 43. Let f ∈ L2(Rn). Consider the following operators

Tyf(x) = f(x− y), translation operatorMkf(x) = eix·kf(x), modulation operatorDsf(x) = |s|−n/2f(s−1x), s ∈ R \ 0 dilation operator


where x · k = k1x1 + · · ·+ xnkn.(i) Find ‖Tyf‖, ‖Mk‖, ‖Dsf‖, where ‖ ‖ denotes the norm in L2(Rn).(ii) Find the adjoint operators of these three operators.

Problem 44. Consider the vector space

H1(a, b) := f(x) ∈ L2(a, b) : f ′(x) ∈ L2(a, b)

with the norm g ∈ H1(a, b))

‖g‖1 :=√‖g‖20 + ‖∂g/∂x‖20.

Consider the Hilbert space L2(−π, π) and f(x) = sin(x). Find the norm‖f‖1.

Problem 45. Let f ∈ H1(a, b). Then for a ≤ x < y ≤ b we have

f(y) = f(x) +∫ y

x

f ′(s)ds.

(i) Show that f ∈ C[a, b].(ii) Show that

|f(y)− f(x)| ≤ ‖f‖1√|y − x|.

Problem 46. Consider the Hilbert space L2[0,∞). The Laguerre poly-nomials are defined by

Ln(x) = exdn

dxn(xne−x), n = 0, 1, 2, . . . .

The first five Laguerre polynomials are given by

L0(x) = 1L1(x) = 1− xL2(x) = 2− 4x+ x2

L3(x) = 6− 18x+ 9x2 − x3

L4(x) = 24− 96x+ 72x2 − 16x3 + x4.

Show that the function

φn(x) =1n!e−x/2Ln(x)

form an orthonormal system in the Hilbert space L2[0,∞).


Problem 47. Consider the Hilbert space L2[−π, π]. A basis in thisHilbert space is given by

B =

1√2πeikx : k ∈ Z

.

Find the Fourier expansion of

f(x) = 1.

Problem 48. Consider the Hilbert space L2[0, 1]. Let Pn be the n + 1-dimensional real linear space of all polynomial of maximal degree n in thevariable x, i.e.

Pn = span 1, x, x2, . . . , xn .

The linear space Pn can be spanned by various systems of basis functions.An important basis is formed by the Bernstein polynomials

Bn0 (x), Bn1 (x), . . . , Bnn(x)

of degree n with

Bni (x) := xi(1− x)n−i, i = 0, 1, . . . , n.

The Bernstein polynomials have a unique dual basis

Dn0 (x), Dn

1 (x), . . . , Dnn(x)

which consists of the n+ 1 dual basis functions

Dni (x) =

n∑j=0

cijBnj (x).

The dual basis functions satisfy

〈Dni (x), Bnj (x)〉 = δij .

(i) Calculate the scalar product

〈Bmi (x), Bnj (x)〉.

(ii) Find the coefficients cij .

Problem 49. Consider Fourier series and analytic (harmonic) functionson the disc

D := z ∈ C : |z| ≤ 1 .


A Fourier series can be viewed as the boundary values of a Laurent series

∞∑n=−∞

cnzn.

Suppose we are given a function f on T. Find the harmonic extension u off into D. This means

∆u = 0 and u = f on ∂D = T

where ∆ := ∂2/∂x2 + ∂2/∂y2.

Problem 50. Consider the compact abelian Lie group U(1)

U(1) = e2πiθ : 0 ≤ θ < 1 .

The Hilbert space L2(U(1)) is the space L2([0, 1]) consisting of all measure-able funcrions f(θ) with period 1 such that∫ 1

0

|f(θ)|2dθ <∞.

Now the set of functions

e2πimθ : m ∈ Z

form an orthonormal basis for the Hilbert space L2([0, 1]). Thus everyf ∈ L2([0, 1]) can be expressed uniquely as

f(θ) =+∞∑

m=−∞cme

2πimθ, cm =∫ 1

0

f(θ)e−2πimθdθ.

Calculate ∫ 1

0

|f(θ)|2.

Problem 51. The Hilbert space L2(R) is the vector space of measur-able functions defined almost everywhere on R such that |f |2 is integrable.H1(R) is the vector space of functions with first derivatives in L2(R). Givetwo examples of such a function.

Problem 52. Consider the Hilbert space L2[−π, π]. The set of functions1√2πe−inx

n∈Z


is an orthonormal basis for L2[−π, π]. Let

K(x, t) =1√2πeitx.

For t fixed find the Fourier expansion of this function.

Problem 53. Consider the vector space C([0, 1]) of continouos functions.We define the triangle function

Λ(x) :=

2x 0 ≤ x ≤ 1/22− 2x 1/2 < x ≤ 1 .

Let Λ0(x) := x andΛn(x) := Λ(2jx− k)

where j = 0, 1, 2, . . ., n = 2j + k and 0 ≤ k < 2j . The functions

1, Λ0, Λ1, . . .

are the Schauder basis for the vector space C([0, 1]). Let f ∈ C([0, 1]).Then

f(x) = a+ bx+∞∑n=1

cnΛn(x).

(i) Find the Schauder coefficients a, b, cn.(ii) Consider g : [0, 1]→ [0, 1]

g(x) = 4x(1− x).

Find the Schauder coefficients for this function.

Problem 54. Let s be a nonnegative integer. Let x ∈ R and hn (n =0, 1, 2, . . . be

hn(x) =(−1)n

2n/2√n! 4√π

exp(x2/2)dne−x

2

dxn.

Thus hn for an orthonormal basis in the Hilbert space L2(R). Consider thesequence

fs(x) =1√s+ 1

s∑n=0

einθhn(x)

where s = 0, 1, 2, . . .. Show that the sequence converges weakly but notstrongly to 0.

Problem 55. Consider the Hilbert space L2[0, 2π) with the scalar prod-uct

〈f1, f2〉 =1

2π

∫ 2π

0

f1(eiθ)f2(eiθ)dθ.


(i) Let f1(z) = z and f2(z) = z2. Find 〈f1, f2〉.(ii) Let f1(z) = z2 and f2(z) = sin(z). Find 〈f1, f2〉.

Problem 56. Consider the Hilbert space L2(R2) with the basis

ψmn(x1, x2) = NHm(x1)Hn(x2)e−(x21+x2

2)/2

where m,n = 0, 1, . . . and N is the normalization factor. Consider thetwo-dimensional potential

V (x1, x2) =a

4(x4

1 + x42) + cx1x2.

(i) Find all linear transformation T : R2 → R2 such that

V (Tx) = V (x).

(ii) Show that these 2× 2 matrices form a group. Is the group abelian.(iii) Find the conjugacy classes and the irreducible representations.(iv) Consider the Hilbert space L2(R2) with the orthogonal basis

ψmn(x1, x2) = Hm(x1)e−x21/2Hn(x2)e−x

22/2

where m,n = 0, 1, 2, . . .. Find the invariant subspaces from the projectionoperators of the irreducible representations.

Problem 57. Consider the Hilbert space L2[0, 1]. Let Pn be the n + 1-dimensional real linear space of all polynomial of maximal degree n in thevariable x, i.e.

Pn = span 1, x, x2, . . . , xn .

The linear space Pn can be spanned by various systems of basis functions.An important basis is formed by the Bernstein polynomials

Bn0 (x), Bn1 (x), . . . , Bnn(x)

of degree n with

Bni (x) := xi(1− x)n−i, i = 0, 1, . . . , n.

The Bernstein polynomials have a unique dual basis D(0x), Dn

1 (x), . . . , Dnn(x)

which consists of the n+ 1 dual basis functions

Dni (x) =

n∑j=0

cijBnj (x).


The dual basis functions satisfy

〈Dni (x), Bnj (x)〉 = δij .

(i) Find the scalar product

〈Bmi (x), Bnj (x)〉.

(ii) Find the coefficients cij .

Problem 58. Let V be a metric vector space. A reproducing kernelHilbert space on V is a Hilbert space H of functions on V such that for eachx ∈ V , the point evaluation functional

δx(f) := f(x), f ∈ H

is continouos. A reproducing kernel Hilbert space H possesses a uniquereproducing kernel K which is a function on V × V characterized by theproperties that for all f ∈ H and x ∈ V , K(x, ·) ∈ H and

f(x) = 〈f,K(x, ·)〉H

where 〈·, ·〉H denotes the inner product on H. The reproducing kernel Kuniquely determines the reproducing kernel Hilbert space H. The repro-ducing kernel Hilbert space of a reproducing kernel K is denoted by HK .The Paley-Wiener space is defined by

S := f ∈ C(Rd) ∩ L2(Rd) : suppf ⊆ [−π, π]d

is a reproducing kernel Hilbert space. The Fourier transform of f ∈ L1(Rd)is given by

f(k) :=1

(√

2π)2d

∫R2d

f(x)e−ix·kdx, k ∈ Rd

where x ·k = x1k1 + · · ·+xdkd is the inner product in Rd. The norm on thevector space S inherits from that in L2(Rd). Show that the reproducingkernel for the Paley-Wiener space S is the sinc function

sinc(x,y) :=d∏j=1

sin(π(xj − yj))π(xj − yj)

, x,y ∈ Rd.

Problem 59. Show that (Mehler’s formula)

exp(−(u2+v2−2uvz)/(1−z2)) = (1−z2)1/2 exp(−(u2+v2))∞∑n=0

zn

n!Hn(u)Hn(v)


where Hn are the Hermite polynomials.

Problem 60. Consider the Hilbert space L2(R) and the one-dimensionalSchrodinger equation (eigenvalue equation)(

− d2

dx2+ V (x)

)u(x) = Eu(x)

where the potential V is given by

V (x) = x2 +ax2

1 + bx2

where b > 0. Insert the ansatz

u(x) = e−x2/2v(x)

and find the differential equation for v. Discuss. Make a polynomial ansatzfor v.

Problem 61. Consider the Hilbert space L2(R). Let g > 0. Considerthe one-dimensional Schrodinger equation (eigenvalue equation)(

− d2

dx2+ x2 +

λx2

1 + gx2

)u(x) = Eu(x).

Find a solution of the second order differential equation by making theansatz

u(x) = A(1 + gx2) exp(−x2/2).

Problem 62. (i) Consider the Hilbert space L2[−1/2, 1/2]. Show thatthe following sets

B1 := φk(x) = exp(2πikx), k ∈ Z B2 := ψk(x) =

√2 sin(2πkx), k ∈ N

each form an orthonormal basis in this Hilbert space.(ii) Expand the step function

f(x) =−1 for x ∈ [−1/2, 0]1 for x ∈ [0, 1/2]

with respect to the basis B1 and with respect to the basis B2. Show thatthe two expansions are equivalent. Recall that

2 sin(x) sin(y) ≡ cos(x− y)− cos(x+ y).


Problem 63. Consider the problem of a free particle in a one-dimensionalbox [−a, a]. The underlying Hilbert space is L2[−a, a]. An orthonormalbasis in L2[−a, a] is given by

B = u(+)k (q), u(−)

k (q) : k ∈ N

where

u(+)k =

1√a

cos(

(k − 1/2)πqa

), u

(−)k =

1√a

sin(kπq

a

).

The formal solution of the initial value problem of the Schrodinger equation

i~∂ψ

∂t= Hψ

is given byψ(t) = exp(−iHt/~)ψ(0).

Letψ(q, 0) =

1√a

sin(πq/a), φ(q) =1√a

sin(πq/a).

Find exp(−iHt/~) and P = |〈φ, ψ(t)〉|2.

Problem 64. Let n be a positive integer. Consider the Hilbert spaceL2[0, n] and the function

f(x) = e−x.

Find a, b ∈ R such that

‖f(x)− (ax2 + bx)‖

is a minimum. The norm in the Hilbert space L2[0, n] is induced by thescalar product.

Problem 65. Give a function f ∈ L2([0,∞)) such that∫ ∞0

f(x)dx = 1,∫ ∞

0

xf(x)dx = 1.

Problem 66. Consider the Hilbert space L2[0, 2π]. The linear operatorLf(x) := df(x)/dx acts on a dense subset of L2[0, 2π]. Show that this linearoperator is not bounded.

Problem 67. Consider the Hilbert space L2(R3, dx) and let

S2 = (x1, x2, x3) : x21 + x2

2 + x23 = 1 .


In spherical coordinates this Hilbert space has the decomposition

L2(R3, dx) = L2(R+, r2dr)⊗ L2(S2, sin(θ)dθdφ).

Let I be the identity operator in the Hilbert space L2(S2, sin(θ)dθdφ). Thenthe radial momentum operator

Pr := −i~1r

(∂

∂r

)r

is identified with the closure of the operator Pr ⊗ I defined on D(Pr) ⊗L2(S2, sin(θ)dθdφ) where

D(Pr) =f ∈ L2(R+, r2dr) : f ∈ AC(R+),

1r

d

drrf(r) ∈ L2(R+, r2dr) lim

r→0r|f(r)| = 0

and for each f ∈ D(Pr)

Prf(r) = −i~1r

d

dr(rf(r))

where Pr is maximal symmetric in L2(R+, r2dr). Show that Pr is not self-adjoint.

Problem 68. Consider the Hilbert space L2[−π, π] and the functions

f(x) = | sin(x)| g(x) = | cos(x)|.

Find the distance‖f(x)− g(x)‖

in this Hilbert space.

Problem 69. Consider the Hilbert space L2(R). Show that the spectrumof the position operator x is the real line denoted by R.

Problem 70. Consider the Hilbert space L2(R). Is

φn(x) =1√

π(1 + x2)e2in arctan(x), n ∈ Z

an orthonormal basis in L2(R)?

Problem 71. Consider the Hilbert space L2[0,∞). Show that the func-tions

φn(x) = e−x/2Ln(x), n = 0, 1, 2, . . .


form an orthonormal basis in L2[0,∞), where Ln are the Laguerre polyno-mials defined by

Ln(x) =x

n!dn

dxn(xne−x) =

n∑k=0

(−1)k

k!

(n

k

)xk.

Problem 72. Consider the Hilbert space L2(R). Show that the functions

φn(x) =1

2n/2√n!(π)1/4

Hn(x)e−x2/2, n = 0, 1, 2, . . .

form an orthonormal basis in the Hilbert space L2(R), where Hn are theHermite polynomials

Hn(x) = (−1)nex2 dne−x

2

dxn, n = 0, 1, 2, . . .

Problem 73. Let b > a and n = 1, 2, . . .. Consider the Hilbert spaceL2[a, b]. Find ∫ b

a

sin2

(nπ(x− a)b− a

).

The functions

φn(x) =

√2

b− asin(nπ(x− a)b− a

)form an orthonormal basis in the Hilbert space L2[a, b].

Problem 74. Consider the Hilbert space L2[0,∞) and

f(x) =

√2π

∫ ∞0

g(y) cos(yx)dy, g(x) =

√2π

∫ ∞0

f(y) cos(yx)dy.

Let g : R+ → Rg(y) = e−y

Find f(x).

Problem 75. Consider the Hilbert space L2[−1, 1]. An orthonormalbasis in this Hilbert space is given by

B =

1√2πeikx : |x| ≤ π, k ∈ Z

.


Consider the function f(x) = eiax in this Hilbert space, where the constanta is real but not an integer. Apply Parseval’s relation

‖f‖2 =∑k∈Z|〈f, φk〉|2, φk(x) =

1√2eikx

to show that∞∑

k=−∞

1(a− k)2

=π2

sin2(ax).

Problem 76. Consider the function f : R→ R

f(x) =x

sinh(x)

with f(0) = 1. Show that

x

sinh(x)= 1 + 2

∞∑j=1

(−1)jx2

x2 + (jπ)2.

Problem 77. Let µ > 0 and

R =

x1

x2

x3

, R′ =

x′1x′2x′3

.

(i) Show thatexp(−|R−R′|/µ)

|R−R′|=

∞∑n=0

εn cos(n(φ−φ′))∫ ∞

0

1k2 + 1/µ

Jn(kr)J(kr′) exp(−√k2 + 1/µ2|x3−x′3|)kdk

where εn = 1 for n = 0 and εn = 2 for n > 0 and Jm(kr) is ordinary Besselfunction of order m.(ii) Consider the functions

fs,k,n(R) =

√k

2πJn(kr)einφ+isx3

where 0 < k <∞, −∞ < s <∞, and n = 0,±1,±2, . . .. Show that∫ 2π

0

∫ +∞

−∞dx3

∫ ∞0

fs,k,n(R)fs′,k′,n′(R)rdr = δnn′δ(s− s′)δ(k − k′)


Problem 78. Let a > 0. Consider the Hilbert space L2([0, a]) and thefunction fn ∈ L2([0, a])

fn(x) =1√ae2πixn/a, n = 1, 2, . . .

Find ‖fn(x)‖ and ‖dfn(x)dx ‖.

Problem 79. Let a > 0. Consider the Hilbert space L2([0, a]) and thelinear bounded operator

Tf(x) = xf(x), f ∈ L2([0, a]).

Find ‖A‖.

Problem 80. Consider the Hilbert space L2([0, 2π]). For any 2π periodicfunction k(τ) in L2([0, 2π]) we define

K(u) :=∫ 2π

0

k(x− τ)u(τ)dτ.

Find the eigenfunctions and eigenvalues of K. Note that the functionseinx : n ∈ Z form an orthonormal basis in L2([0, 2π]).

Problem 81. Let b > a. Consider the Hilbert space L2([a, b]) and thesecond order ordinary differential equations

d2u

dx2+ λu = 0

with the boundary conditions u(a) = u(b) = 0. Solve the differential equa-tion with this boundary condition. Discuss.

Problem 82. Let a, b > 0. Consider the linear partial differential equa-tion

∂2u

∂x41

+ 2∂4u

∂x21∂x

22

+∂4u

∂x42

+ c2∂2u

∂t2= 0.

For the space coordinates x1, x2 we have the domain 0 ≤ x1 ≤ a, 0 ≤ x2 ≤ band the boundary conditions that u(0) = u(a) = u(b) = 0. We consider theHilbert space L2([0, a]× [0, b]) for the space coordinates. Find a solution ofthe partial differential equation.

Problem 83. The Anosov map is defined as follows: Ω = [0, 1)2,

φ(x, y) = (x+ y, x+ 2y).


In matrix form we have(xy

)7→(

1 11 2

)(xy

)mod 1.

(i) Show that the map preserves Lebesgue measure.(ii) Show that φ is invertible. Show that the entire sequence can be recov-ered from one term.(iii) Show that φ is mixing utilizing the Hilbert space L2([0, 1]× [0, 1]).

Problem 84. Let Ω be the unit disk. A Hilbert space of analytic functionscan be defined by

H :=

f(z) analytic, |z| < 1 : sup

a<1

∫|z|=a

|f(z)|2ds <∞

and the scalar product

〈f, g〉 := lima→1

∫|z|=a

f(z)g(z)ds.

Let cn (n = 0, 1, 2, . . .) be the coefficients of the power-series expansion ofthe analytic function f . Find the norm of f .

Problem 85. Let Cn denote the complex Euclidean space. Let z =(z1, . . . , zn) ∈ Cn and w = (w1, . . . , wn) ∈ Cn then the scalar product(inner product) is given by

z ·w := zw∗ = zwT

where z = (z1, . . . , zn). Let En denote the set of entire functions in Cn.Let Fn denote the set of f ∈ En such that

‖f‖2 :=1πn

∫Cn

|f(z)|2 exp(−|z|2)dV

is finite. Here dV is the volume element (Lebesgue mesure)

dV =n∏j=1

dxjdyj =n∏j=1

rjdrjdθj

with zj = rjeiθj . The norm follows from the scalar product of two functions

f, g ∈ Fn〈f, g〉 :=

1πn

∫Cn

f(z)g(z) exp(−|z|2)dV.


Letzm := zm1

1 · · · zmnn

where the multiindex m is defined by m! = m1! · · ·mn! and |m| =∑nj=1mj .

Find the scalar product〈zm, zp〉.

Problem 86. Let Ψ be a complex-valued differentiable function of φ inthe interval [0, 2π] and Ψ(0) = Ψ(2π), i.e. Ψ is an element of the Hilbertspace L2([0, 2π]). Assume that (normalization condition)∫ 2π

0

Ψ∗(φ)Ψ(φ)dφ = 1.

Calculate

=~i

∫ 2π

0

Ψ∗(φ)φd

dφΨ(φ)dφ

where = denotes the imaginary part.

Problem 87. The Fock space F is the Hilbert space of entire functionswith inner product given by

〈f |g〉 :=1π

∫Cf(z)g(z)e−|z|

2dxdy, z = x+ iy

where C denotes the complex numbers. Therefore the growth of functionsin the Hilbert space F is dominated by exp(|z|2/2). Let f, g ∈ F withTaylor expansions

f(z) =∞∑j=0

ajzj , g(z) =

∞∑j=0

bjzj .

(i) Find 〈f |g〉 and ‖f‖2.(ii) Consider the special that f(z) = sin(z) and g(z) = cos(z). Calculate〈f |g〉.(iii) Let

K(z, w) := ezw, z, w ∈ C.

Calculate 〈f(z)|K(z, w)〉.

Problem 88. Let Cn×N be the vector space of all n×N complex matrices.Let Z ∈ Cn×N . Then Z∗ ≡ ZT , where T denotes transpose. One defines aGaussian measure µ on Cn×N by

dµ(Z) :=1

πnNexp(−tr(ZZ∗))dZ


where dZ denotes the Lebesgue measure on Cn×N . The Fock space F(Cn×N )consists of all entire functions on Cn×N which are square integrable withrespect to the Gaussian measure dµ(Z). With the scalar product

〈f |g〉 :=∫

Cn×N

f(Z)g(Z)dµ(Z), f, g ∈ F(Cn×N )

one has a Hilbert space. Show that this Hilbert space has a reproducingkernel K. This means a continuous function K(Z,Z ′) : Cn×N ×Cn×N → Csuch that

f(Z) =∫

Cn×N

K(Z,Z ′)f(Z ′)dµ(Z ′)

for all Z ∈ Cn×N and f ∈ F(Cn×N ).

Problem 89. Consider the Lie group

SU(1, 1) =(

α ββ α

)|α|2 − |β|2 = 1

.

The elements of this Lie group act as analytic automorphism of the disk

Ω := |z| < 1

underz → zg =

αz + β

βz + α

where (zg)h = z(gh). Let n ≥ 2. We define

Hn := f(z) analytic in Ω, ‖f‖2 =∫

Ω

|f(z)|2(1− |z|2)n−2dxdy <∞

andUn(g)f(z) :=

1(βz + α)n

f((αz + β)/(βz + α)).

Then Hn is a Hilbert space, i.e. the analytic functions in

L2(Ω, (1− |z|2)n−2dxdy)

form a closed subspace. Un is a representation, i.e.,

Un(gh) = Un(g)Un(h)

and Un(e) = I, where e is the identity element in SU(1, 1) (2 × 2 unitmatrix).Show that

1(1− |z|2)2

dx ∧ dy

is invariant z → zg.



Problem 1. Consider the Hilbert space L2[0,∞) and the function f ∈L2[0,∞)

f(x) = exp(−u1/4) sin(u1/4).

Show that

〈f, xn〉 =∫ ∞

0

f(x)xndx = 0 for n = 0, 1, 2, . . . .

Problem 2. Let 0 ≤ r < 1. Consider the Hilbert space L2[0, 2π] andf(θ) ∈ L2[0, 2π]. Show that

12π

∫ 2π

0

f(θ)dθ +1π

∫ 2π

0

∞∑j=1

rjf(θ) cos(j(φ− θ))dθ

=1

2π

∫ 2π

0

f(θ)1− r2

1− 2r cos(φ− θ) + r2dθ.

Problem 3. Consider the Hilbert space L2[−π, π]. Find the series

f(θ) =∞∑

n=−∞cne

inθ

where θ ∈ [−π, π] and

c0 = 1, c1 = c−1 = 1, c2 = c−2 =12, . . . , cn = c−n =

1n.

Problem 4. Consider the real axis R and a cover

R = ∪j=∞j=−∞Ij , Ij = [εj , εj+1), εj < εj+1

with dj := εj+1 − εj = |Ij |. Let fj be a window function supported in theinterval [εj − dj−1/2, εj+1 + dj+1/2) such that

j=∞∑j=−∞

f2j (x) = 1

and f2j (x) = 1− f2

j (2εj+1 − x) for x near εj+1. Show that the functions

gj,k(x) =2√2dj

fj(x) sin(π

2dj(2k + 1)(x− εj)

), k = 0, 1, 2, . . .


form an orthonormal basis of the Hilbert space L2(R) subordinate to thepartition fj .

Problem 5. Let

S = L2,real(R, dx/(1 + |x|3))

and q ∈ S. Consider the operator

L(q) := − d2

dx2+ q.

Show that the operator L(q) defines a selfadjoint operator in the Hilbertspace L2(R, dx).

Problem 6. Let a > 0. Consider the Hilbert space L2([0, a] × [0, a] ×[0, a]). Let nj ∈ Z with j = 1, 2, 3. Show that the functions

φn1,n2,n3(x) =1a3/2

exp(i(n1x1 + n2x2 + n3x3))

form an orthonormal basis in this Hilbert space.

Problem 7. Consider the Hilbert space L2[0, 2π], the curve in the planeexpressed in polar coordinates

r(θ) = 1 + r cos(2θ)

and

r(θ) =+∞∑

n=−∞Cn exp(inθ), Cn =

12π

∫ 2π

0

r(θ) exp(−inθ)dθ.

Find the coefficents Cn.

Problem 8. Let `1 > 0, `2 > 0, `3 > 0 with dimension length. Considerthe Hilbert space L2([0, `1]× [0, `2]× [0, `3]). Let n1, n2, n3 ∈ N0.(i) Show that the functions

u(x1, x2, x3) = cos(k1x1) sin(k2x2) sin(k3x3)v(x1, x2, x3) = sin(k1x1) cos(k2x2) sin(k3x3)w(x1, x2, x3) = sin(k1x1) sin(k2x2) cos(k3x3)

in this Hilbert space, where

k1 =n1π

`1, k2 =

n2π

`2, k3 =

n3π

`3.


(ii) Show that u(x1, x2, x3), v(x1, x2, x3), w(x1, x2, x3) are 0 at the bound-aries of the box (rectangular cavity) [0, `1]× [0, `2]× [0, `3].(iii) Show that the functions (A,B,C are constants)

u(x1, x2, x3, t) =Au(x1, x2, x3)eiωt

v(x1, x2, x3, t) =Bv(x1, x2, x3)eiωt

w(x1, x2, x3, t) =Cw(x1, x2, x3)eiωt.

satisfy the wave equation

1c2∂2ψ

∂t2=∂2ψ

∂x21

+∂2ψ

∂x22

+∂2ψ

∂x23

with (dispersion relation))

ωk = c(k21 + k2

2 + k23)

Problem 9. Let ` = 0, 1, 2, . . .. Show that

+∑m=−`

Y`m(θ, φ)Y ∗`m(θ, φ) =2`+ 1

4π.

Problem 10. Consider the Hilbert space L2(R). Let T be the linearoperator of pointwise multiplication on L2(R) given by

(Tf)x = xf(x) for f ∈ L2(R).

Find the spectrum of T .

Problem 11. Let Ai(x) be the Airy function, dAi(x)/dx be the derivativeand an (n = 1, 2, . . .) be the zeros of the Airy functions. Show that thefunctions

Ai(x+ an)dAi(x = an)/dx

, n = 1, 2, . . .

form an orthonormal basis in the Hilbert space L2([0,∞)).

Problem 12. Let b > a. Consider the Hilbert space L2([a, b]). Do thefunctions (Chebyshev polynomial)

Tn(x) =(b− a)n

22n−1cos(n arccos

(2xb− a

− b+ a

b− a

))form an orthonormal basis in L2([a, b])?


Problem 13. Let a > 0. Consider the Hilbert space L2([0, a]).(i) Show that the functions

φn(x) =1√ae2πinx/a, n ∈ Z

form an orthonormal basis in L2([0, a]).(ii) Show that the functions

1√a,√

2/a cos(2πnx/a),√

2/a sin(2πnx/a), n = 1, 2, 3, . . .

form an orthonormal basis in L2([0, a]).(i) Show that the functions

φn(x) =√

2/a sin(πnx/a), n = 1, 2, 3, . . .

form an orthonormal basis in L2([0, a]).(i) Show that the functions

1/√a,

√2/a cos(πnx/a), n = 1, 2, 3, . . .

form an orthonormal basis in L2([0, a]).

Problem 14. Consider the Hilbert space L2[−π, π]. A basis in thisHilbert space is given by

B =

1√2πeikx : k ∈ Z

.

Find the Fourier expansion of f(x) = 1.

Problem 15. Can one construct an orthonormal basis in the Hilbertspace L2(R) starting from (σ > 0)

e−|x|/σ, xe−|x|/σ, x2e−|x|/σ, x3e−|x|/σ, . . . ?

Problem 16. Consider the Hilbert space L2([−1, 1]) and the functionfn ∈ L2([−1,+1]) (n ∈ N)

fn(x)

1 for −1 ≤ x ≤ 0√

1− nx for 0 ≤ x ≤ 1/n0 for 1/n ≤ x ≤ 1.

Show that‖fn(x)− fm(x)‖ ≤ 1

n+

1m.


Note that if n→∞, m→∞, then

‖fn(x)− fm(x)‖2 → 0.

Problem 17. Starting with the set of polynomials 1, x, x2, . . . , xn, . . . use the Gram-Schmidt procedure the scalar product (inner product)

〈f, g〉 =∫ 1

−1

(f(x)g(x) + f ′(x)g′(x))dx

to find the first five orthogonal polynomials, where f ′ denotes derivative.

Problem 18. Let I = [0, 1), µ the Lebesgue measure, f0(x) = 1 and forn ≥ 1

fn(x) =

+1 for 2n−1x = y ∈ [0, 1/2) (mod 1)−1 for 2n−1x = y ∈ [1/2, 1) (mod 1)

Show that the functions fn : [0, 1) → R form an orthonormal sequence inthe Hilbert space L2[0, 1). The functions are called Rademacher functions.

Problem 19. Apply Parseval’s equality to show that

π2

8=∞∑j=1

1(2j − 1)2

,π6

945=∞∑j=1

1j6.

Problem 20. Consider the Hilbert space L2([0, 1]). Let n ≥ 2 andconsider the function fn : [0, 1]→ R

fn(x) =

n2x for 0 ≤ x ≤ 1/n−n2(x− 2/n) for 1/n ≤ x ≤ 2/n

0 for 2/n ≤ x ≤ 1.

Show that ∫ 1

0

fn(x)dx = 1.

Studylimn→∞

fn.

Chapter 4

Hilbert Space `2(N)

4.1 Solved Problems

Problem 1. Consider the Hilbert space `2(N). Let x = (x1, x2, . . .)T bean element of `2(N). We define the linear operator A in `2(N) as

Ax = (x2, x3, . . .)T

i.e. x1 is omitted and the n+1st coordinate replaces the nth for n = 1, 2, . . ..Then for the domain we have D(A) = `2(N). Find A∗y and the domain ofA∗, where y = (y1, y, . . .). Is A unitary?

Problem 2. Consider the Hilbert space `2(N) and x = (x1, x2, . . .)T . Thelinear bounded operator A is defined by

A(x1, x2, x3, . . . , x2n, x2n+1, . . .)T =

(x2, x4, x1, x6, x3, x8, x5, . . . , x2n+2, x2n−1, . . .)T .

Show that the operator A is unitary. Show that the point spectrum of A isempty and the continuous spectrum is the entire unit circle in the λ-plane.

Problem 3. We haveA∗ =

54

Hilbert Space `2(N) 55

Problem 4. Consider the Hilbert space `2(N). Suppose that S and Tare the right and left shift linear operators on this sequence space, definedby

S(x1, x2, . . .) = (0, x1, x2, . . .), T (x1, x2, x3, . . .) = (x2, x3, x4, . . .).

Show that T = S∗.

Problem 5. Find the spectrum of the infinite dimensional matrix

A =

0 1 0 0 0 . . .1 0 1 0 0 . . .0 1 0 1 0 . . .0 0 1 0 1 . . .

. . . . . .

.

In other words

aij =

1 if i = j + 11 if i = j − 10 otherwise

Problem 6. Let Pj (j = 0, 1, 2, . . .) be the Legendre polynomials

P0(x) = 1, P1(x) = x, P2(x) =12

(3x2 − 1), . . . .

Calculate the infinite dimensional matrix A = (Ajk)

Ajk =∫ +1

−1

Pj(x)dPk(x)dx

dx

where j, k = 0, 1, . . .. Consider the matrix A as a linear operator in theHilbert space `2(N0). Is ‖A‖ <∞?

Problem 7. Let Z be the set of integers. Consider the Hilbert space`2(Z2). Let (m1,m2) ∈ Z2. Let f(m1,m2) be an element of `2(Z2). Con-sider the unitary operators

Uf(m1,m2) := e−2πiαm2f(m1 + 1,m2),

V f(m1,m2) := e−2πiβm1f(m1,m2 + 1).

They are the so-called magnetic translation operators with phase α and β,respectively. Find the spectrum of U and V . Find the commutator [U, V ].The so-called Harper operator which is self-adjoint is defined by

H := U + U∗ + V + V ∗.


Find the spectrum of H. Consider the case α, β irrational and α, β rational.

Problem 8. The spectrum σ(H) of a linear operator H is defined as theset of all λ for which the resolvent

R(λ) = (λI − H)−1

does not exist. If the linear operator H is self-adjoint, the spectrum is asubset of the real axis. The Lebesgue decomposition theorem states that

σ = σpp ∪ σac ∪ σsing

where σpp is the countable union of points (the pure point spectrum), σacis absolutely continuous with respect to Lebesgue measure and σsing issingular with respect to Lebesgue measure, i.e. it is supported on a set ofmeasure zero. Consider the Hilbert space `2(Z) and the linear operator

H = · · · ⊗ I2 ⊗ I2 ⊗ σ3 ⊗ σ1 ⊗ σ3 ⊗ I2 ⊗ I2 ⊗ · · ·

where σ1 is at position 0. Find the spectrum of this linear operator.

Problem 9. Let Ω be the unit disk. A Hilbert space of analytic functionscan be defined by

H :=

f(z) analytic |z| < 1 : sup

a<1

∫|z|=a

|f(z)|2ds <∞

and the scalar product

〈f, g〉 := lima→1

∫|z|=a

f(z)g(z)ds.

Let cn (n = 0, 1, 2, . . .) be the coefficients of the power-series expansion ofthe analytic function f . Find the norm of f .

Problem 10. Let |n〉 be the number states (n = 0, 1, . . .). Let k =0, 1, . . .. Define the operators

Tk :=∞∑n=0

|n〉〈2n+ k|.

(i) Show that TkT†k′ = δkk′I.

(ii) Show that T †kTk = Pk is a projection operator.(iii) Show that

∑∞k=0 Pk = I.

Hilbert Space `2(N) 57

(iv) Is the operator∞∑k=0

Tk ⊗ T †k

unitary?

Problem 11. Let A be a square finite-dimensional matrix over R suchthat

AAT = In. (1)

Show thatATA = In. (2)

Does (2) also hold for infinite dimensional matrices ?


Problem 1. Let M be any n × n matrix. Let x = (x1, x2, . . .)T . Thelinear operator A is defined by

Ax = (w1, w2, . . .)T

where

wj =n∑k=1

Mjkxk, j = 1, 2, . . . , n

wj = xj , j > n

and D(A) = `2(N). Show that A is self-adjoint if the n × n matrix M ishermitian. Show that A is unitary if M is unitary.

Chapter 5

Fourier Transform

5.1 Solved ProblemsProblem 1. Consider the Hilbert space L2(R). Find the Fourier trans-form of the function

f(x) =

1 if −1 ≤ x ≤ 0e−x if x ≥ 00 otherwise

Problem 2. (i) Find the Fourier transform for

fα(x) =α

2exp(−α|x|), α > 0.

Discuss α large and α small.(ii) Calculate ∫ ∞

−∞fα(x)dx.

Problem 3. Find the Fourier transform of the hat function

f(t) =

1− |t| for −1 < t < 10 otherwise

58

Fourier Transform 59

Problem 4. Let f ∈ L2(R) and f ∈ L1(R. Assume that f(x) = f(−x).Can we conclude that f(k) = f(−k)?

Problem 5. Consider the Hilbert space L2(R). Find the Fourier trans-form of

f(x) = e−a|x|, a > 0.

Problem 6. Consider the Hilbert space L2(R). Let a > 0. Define

fa(x) =

12a |x| < a0 |x| > a

Calculate ∫Rfa(x)dx

and the Fourier transform of fa. Discuss the result in dependence of a.

Problem 7. Consider the Hilbert space L2(R). Let

ψ(ω) =

1 if 1/2 ≤ |ω| ≤ 10 otherwise

andφ(ω) = e−α|ω|, α > 0.

(i) Calculate the inverse Fourier transform of ψ(ω) and φ(ω), i.e.

ψ(t) =1

2π

∫Re−iωtψ(ω)

φ(t) =1

2π

∫Re−iωtφ(ω).

(ii) Calculate the scalar product 〈ψ(t)|φ(t)〉 by utilizing the identity

2π〈ψ(t)|φ(t)〉 = 〈ψ(ω)|φ(ω)〉.

Problem 8. Consider the Hilbert space L2(R) and the function f ∈L2(R)

f(x) =

1 if |x| < 10 if |x| ≥ 1

Calculate f ∗ f and verify the convolution theorem

f ∗ f = f f .


Problem 9. Let

f(ω) =

(1− ω2) for |ω| ≤ 10 for |ω| > 1

Find f(t).

Problem 10. Let a > 0. Find the Fourier transform of the functionfa : R→ R

fa(x) =

x/a2 + 1/a for −a ≤ x ≤ 0−x/a2 + 1/a for 0 ≤ x ≤ a

0 otherwise

Problem 11. Let a > 0. Find the Fourier transform of

fa(t) =1√ae−a|t|.

Discuss the cases a large and a small. Is fa ∈ L2(R).

Problem 12. Show that the Fourier transform of the rectangular windowof size N

wn =

1 for 0 ≤ n ≤ N − 10 otherwise

is

W (eiω) =sin(ωN/2)sin(ω/2)

e−iω(N−1)/2.

Problem 13. Consider the Hilbert space L2(R). Let T > 0. Considerthe function in L2(R)

f(t) =A cos(Ωt) for −T < t < T

0 otherwise

where A is a positive constant. Calculate the Fourier transform.

Problem 14. Let σ > 0. Show that the Fourier transform of the Gaussianfunction

gσ(x) =1√2πσ

exp(− x2

2σ2

)is again a Gaussian function

gσ(k) = e−σ2k2/2.

Fourier Transform 61

We have∫∞−∞ gσ(x)dx = 1. Is∫ ∞

−∞gσk

(k)dk = 1 ?

Problem 15. Show that the analytic function f : R→ R

f(x) = sech(πx) ≡ 1cosh(πx)

is an element of L2(R) and L1(R). Find the Fourier transform of the func-tion.

Problem 16. Let a > 0. Find the Fourier transform of

√2πfa(x) +

sin(ax)ax

where fa is the function with 1 for |x| ≤ a and 0 otherwise.

Problem 17. Consider the Hermite-Gauss functions

fn(x) =21/4

√2nn!

Hn(√

2πx) exp(−πx2), n = 0, 1, 2, . . .

where Hn is the nth Hermite polynomial. They for an orthonormal basisin the Hilbert space L2(R). Do the Fourier transform of the functions forman orthonormal basis in the Hilbert space L2(R)?

Problem 18. Let ω0 > 0 be a fixed frequency and t the time. Calculate

f(ω) =1√2π

∫ ∞−∞

e−|ω0t|e−iωtdt.


Problem 1. The Hilbert transform h(t) of the function f(t) is the prin-cipal value of the convolution of f(t) with the kernel function k(t) = 1/(πt)

h(t) =∫ ∞−∞

f(s)k(t− s)ds =∫ ∞−∞

f(s)1

t− sds.


LetG(ω) =

∫ ∞−∞

g(t) exp(iωt)dt

be the Fourier transform of g. Show that the Hilbert transform can bewritten as

H(ω) = F (ω)K(ω) = −isgn(ω)F (ω).

Problem 2. Let f ∈ L2(R) and let us call the Fourier transformedfunction f . Is f ∈ L2(R)? What is preserved under the Fourier transform?

Chapter 6

Wavelets

6.1 Solved ProblemsProblem 1. Consider the Hilbert space L2[0, 1] and the function f(x) =x2 in this Hilbert space. Project the function f onto the subspace of L2[0, 1]spanned by the functions φ(x), ψ(x), ψ(2x), ψ(2x− 1), where

φ(x) :=

1 for 0 ≤ x < 10 otherwise

ψ(x) :=

1 for 0 ≤ x < 1/2−1 for 1/2 ≤ x < 10 otherwise

.

This is related to the Haar wavelet expansion of f . The function φ is calledthe father wavelet and ψ is called the mother wavelet.

Problem 2. Consider the function H ∈ L2(R)

H(x) =

1 0 ≤ x ≤ 1/2−1 1/2 ≤ x ≤ 10 otherwise

LetHmn(x) := 2−m/2H(2−mx− n)

where m,n ∈ Z. Draw a picture of H11, H21, H12, H22. Show that

〈Hmn(x), Hkl(x)〉 = δmkδnl, k, l ∈ Z

63


where 〈 . 〉 denotes the scalar product in L2(R) Expand the function

f(x) = exp(−|x|)

with respect to Hmn. The functions Hmn form an orthonormal basis inL2(R).

Problem 3. Consider the Hilbert space L2[0, 1] and the Haar scalingfunction (father wavelet)

φ(x) =

1 if 0 ≤ x < 10 otherwise

Let n be a positive integer. We define

gk(x) :=√nφ(nx− k), k = 0, 1, . . . , n− 1.

(i) Show that the set of functions g0, g1, . . . , gn−1 is an orthonormal setin the Hilbert space L2[0, 1].(ii) Let f be a continuous function on the unit interval [0, 1]. Thus f ∈L2[0, 1]. Form the projection fn on the subspace Sn of the Hilbert spaceL2[0, 1] spanned by g0, g1, . . . gn−1 , i.e.

fn =n−1∑k=0

〈f, gk〉gk .

Show that fn(x)→ f(x) pointwise in x as n→∞.

Problem 4. The continuous wavelet transform

Wf(a, b) =1a

∫ +∞

−∞f(t)ψ

(t− ba

)dt, (a, b ∈ R, a > 0)

decomposes the function f ∈ L2(R) hierarchically in terms of elementarycomponents ψ((t−b)/a). They are obtained from a single analyzing waveletψ applying dilations and translations. Here ψ denotes the complex conju-gate of ψ and a is the scale and b the shift parameter. The function ψ hasto be chosen so that it is well localized both in physical and Fourier space.The signal f(t) can be uniquely recovered by the inverse wavelet transform

f(t) =1Cψ

∫ +∞

−∞

∫ +∞

0

Wf(a, b)ψ(t− ba

)da

adb

if ψ(t) (respectively its Fourier transform ψ(ω) satisfies the admissibilitycondition

Cψ =∫ +∞

0

|ψ(ω)|2

ωdω <∞.

Wavelets 65

Consider the analytic function

ψ(t) = te−t2/2.

Does ψ satisfies the admissibility condition?

Problem 5. Consider the function H ∈ L2(R)

H(x) =

1 0 ≤ x < 1/2−1 1/2 ≤ x ≤ 10 otherwise

LetHmn(x) := 2−m/2H(2−mx− n)

where m,n ∈ Z. Draw a picture of H11, H21, H12, H22. Show that

〈Hmn(x), Hkl(x)〉 = δmkδnl, k, l ∈ Z

where 〈 . 〉 denotes the scalar product in L2(R). Expand the function

f(x) = exp(−|x|)

with respect to Hmn. The functions Hmn form an orthonormal basis inL2(R).

Problem 6. Consider the Hilbert space L2(R). Let φ ∈ L2(R) andassume that φ satisfies ∫

Rφ(t)φ(t− k)dt = δ0,k

i.e. the integral equals 1 for k = 0 and vanishes for k = 1, 2, . . .. Show thatfor any fixed integer j the functions

φjk(t) := 2j/2φ(2jt− k), k = 0,±1,±2, . . .

form an orthonormal set.

Problem 7. Consider the function φ : R→ R

ψ(x) :=

1 for x ∈ [0, 1]0 otherwise

Find ψ(x) := φ(2x)− φ(2x− 1). Calculate∫ ∞−∞

ψ(x)dx.


Problem 8. Consider the Littlewood-Paley orthonormal basis of wavelets.The mother wavelet of this set is

L(x) :=1πx

(sin(2πx)− sin(πx)).

Show thatLmn(x) =

12m/2

L(2−mx− n), m, n ∈ Z

generates an orthonormal basis in the Hilbert space L2(R). Apply the ruleof L’Hospital to find L(0).


Problem 1. (i) Consider the Hilbert space L2(R) and φ ∈ L2(R). Thebasic scaling function (father wavelet) satisfies a scaling relation of the form

φ(x) =N−1∑k=0

akφ(2x− k).

Show that the Hilbert transform of φ

H(φ)(y) =1π

∫R

φ(x)x− y

dx

is a solution of the same scaling relation. Note that the scaling function φmay have compact support, the Hilbert transform has support on the realline and decays as y−1.(ii) Show that the Hilbert transform of the related mother wavelet ψ is alsononcompact and decays like y−p−1 where∫

Rxmψ(x)dx = 0

for m = 0, 1, . . . , p− 1.

Problem 2. Isf(x) = e−x

2/2 cos(x)

a mother wavelet for the Hilbert space L2(R).

Problem 3. The linear spline function f : R→ R is defined as

f(x) =

1− |x| −1 ≤ x ≤ 10 otherwise

Wavelets 67

Show that the function generates an orthonormal wavelet basis in theHilbert space L2(R) using the technique of multiresolution analysis.A wavelet basis of the Hilbert space L2(R) is an unconditional basis ofL2(R) generated by the dilations and translations of a wavelet function ψ(mother wavelet)

ψj,k , ψj,k(x) = 2j/2ψ(2jx− k).

Problem 4. Show that for the Shannon wavelet the projection operatorsare given by

(Pj)(x) =∫

R

sin(2jπ(y − x))π(y − x)

f(y)dy.

Chapter 7

Linear Operators

7.1 Solved ProblemsProblem 1. Show that an isometric operator need not be a unitaryoperator.

Problem 2. Consider the Hilbert space L2[0, 1]. Show that the linearoperator T : L2[0, 1]→ L2[0, 1] defined by

Tf(x) = xf(x)

is a bounded self-adjoint linear operator without eigenvalues.

Problem 3. Show that if two bounded self-adjoint linear operators S andT on a Hilbert space H are positive semi-definite and commute (ST = TS),then their product ST is positive semi-definite. We have to show that〈STf, f〉 ≥ 0 for all f ∈ H.

Problem 4. Let a > 0. Consider the Hilbert space L2[−a, a]. Considerthe Hamilton operator

H =~2

2md2

dx2+ V (x)

where

V (x) =

0 for |x| ≤ a∞ otherwise

68

Linear Operators 69

Solve the Schrodinger equation, where the initial function ψ(t = 0) = φ(x)is given by

φ(x) =x/a2 + 1/a for−a ≤ x ≤ 0−x/a2 + 1/a for 0 ≤ x ≤ a

Normalize φ. Calculate the probability to find the particle in the state

χ(x) =1√a

sin(πxa

)after time t. A basis in the Hilbert space L2[−a, a] is given by

1√a

sin(nπx

a

),

1√a

cos(

(n− 1/2)πxa

)n = 1, 2, . . .

.

Problem 5. Show that in one dimensional problems the energy spectrumof the bound state is always non-degenerate. Hint. Suppose that the oppsiteis true. Let u1 and u2 be two linearly independent eigenfunctions with thesame energy eigenvalues E, i.e.

d2u1

dx2+

2m~2

(E − V )u1 = 0,d2u2

dx2+

2m~2

(E − V )u2 = 0.

Problem 6. A particle is enclosed in a rectangular box with imprenetra-ble walls, inside which it can move freely. The Hilbert space is L2([0, a] ×[0, b] × [0, c]). Find the eigenfunctions and eigenvalues. What can be saidabout the degeneracy, if any, of the eigenfunctions.

Problem 7. Conside the Hilbert space L2[0, 1] and the linear operatorT : L2[0, 1]→ L2[0, 1] defined by

(Tf)(x) := xf(x).

Show that T is self-adjoint and positive definite. Find its positive squareroot.

Problem 8. Consider the Hilbert space `2(N) and the linear operator Tdefined by

T : (x1, x2, x3, . . .) 7→ (0, 0, x3, x4, . . .).

Is T bounded? Is T self-adjoint? If so is T positive?

Problem 9. In classical mechanics we have

L = r× p, T = r× F


where T is the torque, F = −∇V (V potential depending only on r) and

dLdt

= T .

In quantum mechanics with p → −i~∇, r → r and wave function ψ wehave

L = −i~∫

R3d3xψ∗(r×∇)ψ

and

T = −∫

R3d3xψ∗(r×∇V )ψ

since F = −∇V . ψ and ψ∗ obey the Schrodinger equation

i~∂ψ

∂t= − ~2

2m∇2ψ + V ψ

−i~∂ψ∗

∂t= − ~2

2m∇2ψ∗ + V ψ∗ .

Show thatdLdt

= T.

Problem 10. Let H be a bounded self-adjoint Hamilton operator withnormalized eigenfunctions φj (j ∈ I) which form an orthonormal basis inthe underlying Hilbert space. We can write

ψ(t) =∑j∈I

cje−iEjt/~φj

where Ej are the eigenvalues of H. Find P (t) = 〈ψ(t = 0)|ψ(t)〉.

Problem 11. Consider the Hamilton operator

H = − ~2

2md2

dx2+D(1− e−αx)2 + eEx cos(ωt)

where α > 0. So the third term is a driving force. Find the quantumLiouville equation for this Hamilton operator.

Problem 12. Consider the Schrodinger equation

i~∂ψ

∂t=(− 1

2m∆ + V (x)

)ψ

Linear Operators 71

Find the coupled system of partial differential equations for

ρ := ψ∗ψ, v := =(∇ψψ

).

Problem 13. Let a > 0. Consider the Hilbert space L2([0, a]) and thelinear bounded operator A defined by

Af(x) := xf(x), f ∈ L2([0, a])

Find ‖A‖.


Problem 1. Consider the Hilbert space L2(R). Let k ∈ Z. For k = 0 wedefine s0 = 0, for k ≥ 1 we define

sk := 1 +12

+13

+ · · ·+ 1k

and for k < 0 we define sk = −s−k. Let ε > 0. Define the indicatorfunctions Wk as

Wk(x) :=

1 for sk < x/ε ≤ sk+1

0 otherwise

Let u ∈ L2(R). Define the linear operator O as

(Ou)(x) := g(x)u(x)

where

g(x) = −xε

+∑k∈Z

(sk + sk+1

2

)Wk(x).

(i) Show that O is a bounded self-adjoint operator for any ε > 0.(ii) Show that the norm of O

‖O‖ = sup‖u‖=1

‖Ou‖

is given by 1/2.


Problem 2. Consider the momentum operator

p = −i ddx

defined on C∞0 [0,∞). Show that the operator has no self-adjoint extensionson the Hilbert space L2[0,∞).

Problem 3. Consider the Hilbert space L2(R) and the linear operator

p = −i ddx

on D = f : f,df

dx∈ L2(R).

Show that the spectrum is the whole real axis.

Problem 4. Consider the Hilbert space L2(R). Let f ∈ L2(R) and θ ∈ R.We define the operator U(θ) as

U(θ)f(x) := eiθ/2f(xeiθ).

Is the operator U(θ) unitary?

Chapter 8

Generalized Functions

8.1 Solved ProblemsProblem 1. Consider the function H : R→ R

H(x) :=

1 0 ≤ x ≤ 1/2−1 1/2 ≤ x ≤ 10 otherwise

Find the derivative of H in the sense of generalized functions. ObviouslyH can be considered as a regular functional∫

RH(x)φ(x)dx.

Find the Fourier transform of H. Draw a picture of the Fourier transform.

Problem 2. Let Cm[a, b] be the vector space of m-times differentiablefunctions and the m-th derivative is continuous over the interval [a, b] (b >a). We define an inner product (scalar product) of such two functions fand g as

〈f, g〉m :=∫ b

a

(fg +

df

dx

dg

dx+ · · ·+ dmf

dxmdmg

dxm

)dx.

Given (Legendre polynomials)

f(x) =12

(3x2 − 1), g(x) =12

(5x3 − 3x)

73


and the interval [−1, 1], i.e. a = −1 and b = 1. Show that f and g areorthogonal with respect to the inner product 〈f, g〉0. Are they orthogonalwith respect to 〈f, g〉1?

Problem 3. Let P be the parity operator, i.e.

Pr := −r.

Obviously, P = P−1. We define

OPu(r) := u(P−1r) ≡ u(−r).

The vector r can be expressed in spherical coordinates as

r = r(sin θ cosφ, sin θ sinφ, cos θ)

where0 ≤ φ < 2π 0 ≤ θ < π.

(i) Calculate P (r, θ, φ).(ii) Let

Ylm(θ, φ) =(−1)l+m

2ll!

(2l + 1

4π(l −m)!(l +m)!

)1/2

(sin θ)mdl+m

d(cos θ)l+m(sin θ)2leimφ

be the spherical harmonics. Find OPYlm.

Problem 4. In the Hilbert space H = `2(N0) Bose annihilation andcreation operators denoted by b and b† are defined as follows: They have acommon domain

D(b) = D(b†) =

ξ = (x0, x1, x2, . . .)T :∞∑j=0

j|xj |2 <∞

.

Then bη is given by

b(x0, x1, x2, . . .)T = (x1,√

2x2,√

3x3, . . .)T

and b†η is given by

b†(x0, x1, x2, . . .) = (0, x0,√

2x1,√

3x2, . . .).

The infinite dimensional vectors

un = (0, 0, . . . , 0, 1, 0, . . .)T

Generalized Functions 75

where the 1 is at the n position (n = 0, 1, 2, . . .) form the standard basis inH = `2(N0). Is

ξ = (1, 1/2, 1/3, . . . , 1/n, . . .)

an element of D(a)?

Problem 5. Given a function (signal) f(t) = f(t1, t2, . . . , tn) ∈ L2(Rn)of n real variables t = (t1, t2, . . . , tn). We define the symplectic tomogramassociated with the square integrable function f

w(X,µ,ν) =n∏k=1

12π|νk|

∣∣∣∣∣∣∫

Rn

dt1dt2 · · · dtnf(t) exp

n∑j=1

(iµj2νj

t2j −iXj

νjtj

)∣∣∣∣∣∣2

where (νj 6= 0 for j = 1, 2, . . . , n)

X = (X1, X2, . . . , Xn), µ = (µ1, µ2, . . . , µn), ν = (ν1, ν2, . . . , νn).

(i) Prove the equality∫Rn

w(X,µ,ν)dX =∫

Rn

|f(t)|2dt (1)

for the special case n = 1. The tomogram is the probabilty distributionfunction of the random variable X. This probability distribution functiondepends on 2n extra real parameters µ and ν.(ii) The map of the function f(t) onto the tomogram w(X,µ,ν) is invert-ible. The square integrable function f(t) can be associated to the densitymatrix

ρf (t, t′) = f(t)f(t′).

This density matrix can be mapped onto the Ville-Wigner function

W (q,p) =∫

Rn

ρf

(q +

u2,q− u

2

)e−ip·udu.

Show that this map is invertible.(iii) How is the tomogram w(X,µ,ν) related to the Ville-Wigner function?(iv) Show that the Ville-Wigner function can be reconstructed from thefunction w(X,µ,ν).(v) Show that the density matrix f(t)f∗(t′) can be found from w(X,µ,ν).

Problem 6. Describe the one-dimensional scattering of a particle incidenton a Dirac delta function, i.e.

U(q) = U0δ(q)


where u0 > 0. Find the transmission and reflection coefficient.

Problem 7. (i) Give the definition of the current density, transmissioncoefficient, and reflection coefficient.(ii) Calculate the transmission and the reflection coefficients of a particlehaving total energy E, at the potential barrier given by

V (x) = aδ(x), a > 0


12π

∞∑k=−∞

eikx =∞∑

k=−∞

δ(x− 2kπ)

in the sense of generalized functionsHint. Expand the 2π periodic function

f(x) =12− x

2π

into a Fourier series.

Problem 9. (i) Give the definition of a generalized function.(ii) Calculate the first and second derivative in the sense of generalizedfunction of

f(x) =

0 x < 04x(1− x) 0 ≤ x ≤ 1

0 x > 1

(iii) Calculate the Fourier transform of f(x) = 1 in the sense of generalizedfunctions.

Problem 10. Consider the generalized function

f(x) = | cos(x)|.

Find the derivative in the sense of generalized functions.

Problem 11. Consider the generalized function

f(x) :=

cos(x) for x ∈ [0, 2π)0 otherwise

Find the first and second derivative of f in the sense of generalized func-tions.


Problem 12. Find the derivative of f : R→ R

f(x) = |x|

in the sense of generalized functions.

Problem 13. Find the first three derivatives of the function f : R → Rdefined by

f(x) = e−|x|


Problem 14. The Sobolev space of order m, denoted by Hm(Ω), is definedto be the space consisting of those functions in the Hilbert space L2(Ω) that,together with all their weak partial derivatives up to and including thoseof order m, belong to the Hilbert space L2(Ω), i.e.

Hm(Ω) := u : Dαu ∈ L2(Ω) for all α such that |α| ≤ m .

We consider real-valued functions only, and make Hm(Ω) an inner productspace by introducing the Sobolev inner product 〈·, ·〉Hm defined by

〈u, v〉Hm :=∫

Ω

∑|α|≤m

(Dαu)(Dαv)dx for u, v ∈ Hm(Ω).

This inner product generates the Sobolev norm ‖ · ‖Hm defined by

‖u‖2Hm = 〈u|u〉Hm =∫

Ω

∑|α|≤m

(Dαu)2dx.

Thus H0(Ω) = L2(Ω). We can write

〈u, v〉 =∑|α|≤m

〈Dα, Dαv〉L2(Ω).

In other words the Sobolev inner product 〈u, v〉Hm(Ω) is equal to the sumof the L2(Ω) inner products of Dαu and Dαv over all α such that |α| ≤ m.(i) Consider the domain Ω = (0, 2) and the function

u(x) =

x2 0 < x ≤ 12x2 − 2x+ 1 1 < x < 2.

Obviously u ∈ L2(Ω). Find the Sobolev space to which u belongs.(ii) Find the norm of u.


Problem 15. Let c > 0. Consider the Schrodinger equation

− ~2

2md2u

dx2+ cδ(n)(x)u = Eu

where δ(n) (n = 0, 1, 2, . . .) denotes the n-th derivative of the delta function.Derive the joining conditions on the wave function u.

Problem 16. Let a > 0. Show that

∞∑m=−∞

exp(i2πm(x+ q)/a) ≡ a∞∑

k=−∞

δ(x+ q − ka).

Problem 17. Let f : R → R be a differentiable function and x0 a rootof f , i.e. f(x0) = 0. Show that

δ′(f(x)) =1

(f ′(x0))2

(δ′(x− x0) +

f ′′(x0)f ′(x0)

δ(x− x0))

Problem 18. Show that the sum

12

∞∑`=0

(2`+ 1)P`(x)P`(y)

of Legendre polynomials P` is given by the Dirac delta function δ(y − x)for −1 ≤ x ≤ +1 and −1 ≤ y ≤ +1.


δ(x− x′) =1

2π+

1π

∞∑n=1

(cos(nx) cos(nx′) + sin(nx) sin(nx′)).


Miscellaneous

Problem 20. Leti~∂ψ

∂t= Hψ

be the Schrodinger equation, where

H = − ~2

2m∆ + U(r), ∆ :=

∂2

∂x21

+∂2

∂x22

+∂2

∂x23

and r = (x1, x2, x3). Let

ρ(r, t) := ψ(r, t)ψ(r, t)

Find j such that

divj +∂ρ

∂t= 0.

Problem 21. A particle is enclosed in a rectangular box with impene-trable walls, inside which in can move freely. The Hilbert space is

L2([0, a]× [0, b]× [0, c])

where a, b, c > 0. Find the eigenfunctions and eigenvalues. What can besaid about the degeneracy, if any, of the eigenfunctions?

Problem 22. Show that in one-dimensional problems the energy spec-trum of the bound states is always non-degenerate.Hint. Suppose that the opposite is true.

Let u1 and u2 be two linearly independent eigenfunctions with the sameenergy eigenvalues E.

d2u1

dx2+

2m~2

(E − V )u1 = 0

d2u2

dx2+

2m~2

(E − V )u2 = 0

Problem 23. Let a > 0 and let fa : R→ R be given by

fa(x) =x/a2 + 1/a for−a ≤ x ≤ 0−x/a2 + 1/a for 0 ≤ x ≤ a

The function fa generates regular functional. Find the derivative of fa inthe sense of generalized functions.


Problem 24. Consider a one-dimensional lattice (chain) with latticeconstant a. Let k be the sum over the first Brioullin zone we have

1N

∑k∈1.BZ

F (ε(k))→ a

2π

∫ π/a

−π/aF (ε(k))dk = G

whereε(k) = ε0 − 2ε1 cos(ka).

Using the identity ∫ ∞−∞

δ(E − ε(k))F (E)dE ≡ F (ε(k))

we can write

G =a

2π

∫ ∞−∞

F (E)

(∫ π/a

−π/aδ(E − ε(k))dk

)dE.

Calculate

g(E) =∫ π/a

−π/aδ(E − ε(k))dk

where g(E) is called the density of states.

Problem 25. Let ε > 0. Consider the Schrodinger eigenvalue equation(− d2

dx2+ 2εδ(x)

)u(x, ε) = E(ε)u(x, ε)

with the boundary conditions u(±1, ε) = 0. Here ε is the coupling con-stant and determines the penetrability of the potential barrier. Find theeigenfunctions and the eigenvalues.

Problem 26. Show that in the sense of generalized functions

δ(x) =12

limε→0

1εe−|x|/ε

δ(x) =1π

limε→∞

εsin2(εx)

(εx)2

δ(x) =14

limε→0

1ε

(1 +|x|ε

)e−|x|/ε.

Problem 27. Give two interpretations of the series of derivatives of δfunctions

f(k) = 2π∞∑n=0

cn(−1)nδ(n)(k). (1)


Problem 28. Let a > 0. Show that

δ(x2 − a2) =12a

(δ(x− a) + δ(x+ a)).

Problem 29. (i) Show that the Fourier transform in the sense of gener-alized function of the Dirac comb∑

n∈Zδ(x− n)

is again a Dirac comb.(ii) Find the Fourier transform in the sense of generalized functions of

1 +√

2πδ(x).

Problem 30. (i) Consider the nonlinear differential equation

3udu

dx= 2

du

dx

d2u

dx2+ u

d3u

dx3.

Show that u(x) = e−|x| is a solution in the sense of generalized function.(ii) Consider the nonlinear partial differential equation

∂u

∂t− ∂3u

∂x2∂t+ 3u

∂u

∂x= 2

∂u

∂x

∂2u

∂x2+ u

∂3u

∂x3.

Show that u(x, t) = c exp(−|x − ct|) (peakon) is a solution in the sense ofgeneralized functions.

Problem 31. Let f be a differentiable function with a simple zero at x =a such that f(x = a) = 0 and df(x = a)/dx 6= 0. Let g be a differentiablefunction with a simple zero at x = b 6= a such that g(x = b) = 0 anddg(x = b)/dx 6= 0. Show that

δ(f(x)g(x)) =1

|f ′(a)g(a)|δ(x− a) +

1|f(b)g′(b)|

δ(x− b)

where ′ denotes differentiation.

Problem 32. Consider the non-relativistic hydrogen atom, where a0 isthe Bohr radius and a = a0/Z. The Schrodinger-Coulomb Green functionG(r1, r2;E) corresponding to the energy variable E is the solution of thepartial differential equation(

− ~2

2m∇2

1 −~2

amr1− E

)G(r1, r2;E) = δ(r1 − r2)


with the appropiate boundary conditions. Show that expanding G in termsof spherical harmonics Y`m

G(r1, r2;E) =∞∑`=0

∑m=−`

g`(r1, r2;E)Y`m(θ1, φ1)Y ∗`m(θ2, φ2)

we find for the radial part g` of the Schrodinger-Coulomb Green function(1r21

d

dr1

(r21

d

dr1

)− `(`+ 1)

r21

+2ar1− 1ν2a2

)g`(r1, r2; ν) = −2m

~2

δ(r1 − r2)r1r2

where ν2a2 := −~2/(2mE).Hint. Utilize the identity

δ(r1 − r2) =δ(r1 − r2)r1r2

∞∑`=0

∑m=−`

Y`m(θ1, φ1)Y ∗`m(θ2, φ2)


δ(t− x

c

)=

12π

∫ ∞−∞

e−iωt(t−x/c)dω.

Problem 34. Let ε > 0. Show that

fε(x− a) =1√πε

exp(− (x− a)2

ε

)tends to δ(x− a) in the sense of generalized function if ε→ 0+.

Problem 35. Let J0 be the Bessel functions. Show that

δ(x)δ(x) =1

2π

∫ ∞0

kJ0(k√

(x2 + y2))dk


Problem 36. Let α ∈ [0, 1). Show that∫ ∞0

xα−1P

(1

1− x2

)=π

2cot(πα/2).


δ(x− x′) = limα→β

(2π

β − α

)3/2

exp(− αβ

2(β − α)(x− x′)2

).


Problem 38. Let p ∈ [0, 1] and

ρ(x) =12pe−|x| + (1− p)δ(x).

Then ρ(x) ≥ 0. Show that in the sense of generalized function∫Rρ(x)dx = 1.

Problem 39. The two-dimensional Dirac comb function is defined by

C(x1, x1) :=∑m∈Z

∑n∈Z

δ(x1 −m)δ(x2 − n).

Find the Fourier transform of C in the sense of generalized functions.

Problem 40. Let T > 0. Consider the sequence of functions

fn(t) =1n!n

T

(nt

T

)nexp(−nt/T )

where n = 1, 2, . . .. Find fn(t) for n → ∞ in the sense of generalizedfunctions. Find the Laplace transform of fn(t).

Problem 41. What charge distribution ρ(r) does the spherical symmetricpotential

V (r) =e−µr

r

give? For r 6= 0 Poisson’s equation in spherical coordinates is given by

∆V (r) =1r

d2

dr2(rV (r)) +R(θ, φ)V (r) = −4πρ(r)

where R(θ, φ) is the differential operator depending on the angles θ, φ.

Problem 42. Let a1, a2, . . . , an 6= 0. Show that

1a1a2 · · · an

= (n− 1)!∫ 1

0

dε1 · · ·∫ 1

0

dεnδ(1−

∑nj=1 εj

(∑nj=1 εjaj)n

.

Problem 43. Prove the identity in the sense of generalized function

f(y)∂

∂yδ(x− y) ≡ −f(x)

∂

∂xδ(x− y) +

df(y)dy

δ(x− y).


Problem 44. Let φ ∈ S(R). Show that in the sense of generalizedfunctions

limζ→∞

sin(ζx)x

φ(x)dx = πφ(0).


u(x, t) =1√

4πDtexp

(− (x− x0)2

4Dt

)satisfies the one-dimensional diffusion equation

∂u

∂t= D

∂2u

∂x2

with the initial condition

u(x, 0) = δ(x− x0).

Problem 46. Let A, B be n× n matrices. Show that

eA+B =∫ ∞

0

dα1eα1Aδ(1− α1) +

∫ ∞0

∫ ∞0

dα1dα2eα1ABeα2Aδ(1− α1 − α2)

+∫ ∞

0

∫ ∞0

∫ ∞0

eα1ABeα2ABeα3Aδ(1− α1 − α2 − α3) + · · ·

Problem 47. Consider the Hilbert space L2([−1, 1]). The Legendre poly-nomials are given by

P0 = 1, Pn(x) =1

2nn!dn

dxn(x2 − 1)n

where n = 1, 2, . . .. They satisfy∫ +1

−1

Pm(x)Pn(x)dx =2

2n+ 1δm,n, n,m = 0, 1, 2, . . .

Let

δ(x) =∞∑j′=0

dj′Pj′(x).

Find the expansion cofficients dj′ .



Problem 1. Show that the 2-dimensional complex δ-function can bewritten as (α ∈ C)

δ(2)(z) =1π2

∫Cd2α exp(α∗z − z∗α) =

1π2

∫Cd2α exp(i(α∗z + z∗α)).


δ(x− x′) =1π

(1 + 2

∞∑k=1

cos(kx) cos(kx′)

).

Problem 3. Consider the Hilbert space L2([0,∞)). The Laguerre poly-nomials are defined as

Ln(x) = exdn

dxn(e−xxn), n = 0, 1, 2, . . .

For the Hilbert space L2([0,∞)) we have the basis

B = e−x/2Ln(x) : n = 0, 1, 2, . . . .

Let a ∈ R. Show that

δ(x− a) = e−(x+a)/2∞∑k=0

Lk(x)Lk(a).


δ(cos(θ1)− cos(θ2))δ(φ1 − φ2) =∞∑`=0

∑m=−`

Y`,m(θ1, φ1)Y ∗`,m(θ2, φ2).

Problem 5. Let Hk (k = 0, 1, 2, . . .) be the Hermite polynomials. Showthat

δ(x− a) =e−(x2+a2)/2

√π

∞∑k=0

Hk(x)Hk(a)2kk!


∂

∂z

(1z

)= πδ(z)


where∂

∂z≡ 1

2

(∂

∂x+ i

∂

∂y

).

Problem 7. Let a ∈ R. Show that

δ(x− a) =∫ +∞

−∞Ai(s− x)Ai(s− a)ds

where Ai is the Airy function.

Problem 8. Let x ∈ R and α > 0. Show that∞∑

n=−∞δ(x− nα) ≡ 1

α

∞∑n=−∞

exp(2πinx/α).

Problem 9. Find the first and second derivative of the function

f(x) :=x exp(−cx2) for x ≥ 0

0 for x < 0

in the sense of generalized function. Note that the function is continuous.

Problem 10. Show that∫ ∞0

cos(ωt)dt = πδ(ω).

Problem 11. Let r2 = x21 + x2

2 and r′2 = x

′21 + x

′22 .

(i) Show that

δ(x1, x2) =1

2πrδ(r)

(ii) Show that

δ(x1 − x′1, x2 − x′2) =1rδ(r − r′)δ(θ − θ′), r′ > 0

where we applied polar coordinates.

Problem 12. Let r2 = x21 + x2

2 + x23. Show that

δ(x1 − x′1, x2 − x′2, x3 − x′3) =1

r sin(θ)δ(r − r′)δ(θ − θ′)δ(φ− φ′)


where we applied spherical coordinates.

Problem 13. Consider the linear partial differential equation

∂u

∂t+ c

∂u

∂x= 0.

Show that u(x, t) = f(x − ct) is a solution for any one-dimensional gener-alized function.

Problem 14. Let p > 0. Show that

12π

∑m∈Z

ei(m+1/2)φ cos(p|m+ 1/2|x) =12p

(δ(x− φ/p) + δ(x+ φ/p)) .

Problem 15. Let a > 0. Consider

fa(x) =

0 for x < −aexp(− exp(x/(x2 − a2))) for−a ≤ x ≤ +a

1 for a < x

Is the function analytic? Is the function an element of C∞(R)? Find

δa(x) =d

dxfa(x).

Then consider a→ 0.

Problem 16. The Morlet wavelet consists of a plane wave modulated bya Gaussian, i.e.

ψ(η) =1

π1/4eiωηe−η

2/2

where ω is the dimensionless frequency. Show that if ω = 6 the admissibilitycondition is satisfied.

Problem 17. Letf0(x) = exp(−x2/2).

We define the mother wavelets fn as

fn(x) = − d

dxfn−1(x), n = 1, 2, . . .

Show that the family of fn’s obey the Hermite recursion relation

fn(x) = xfn−1(x)− (n− 1)fn−2(x), n = 2, 3, . . . .



H(x− a) =∫ ∞−∞

du

∫ ∞−∞

dτ

2πexp(iu(τ − x)).

Problem 19. Show that (distributional identity on L1(R))

1π2

∫R

1(t− x)(s− x)

dx = δ(t− s)

where the integral is evaluated in the principal value sense.

Problem 20. Let c > 0. Show that an integral representation of thedelta function is given by

δ(x) =1

2πi

∫ c+i∞

c−i∞etxdt

where the path of the t-integration can be closed to the right or left.

Problem 21. Show that in the sense of generalized functions

δ(x) =1

2π

∫ +∞

−∞exp(ikx)dx, θ(x) =

∫ ∞0

δ(λ− x)dλ.


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92 Bibliography

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Index

Admissibility condition, 64Airy function, 51Analysing wavelet, 64Anosov map, 45Antilinear operator, 8

Bell basis, 11, 14Bernstein polynomials, 35, 38Bessel function, 44Bessel inequality, 9

Cauchy sequence, 7Chebyshev polynomial, 51Chebyshev polynomials, 32Comb, 8Continuous wavelet transform, 64Convolution theorem, 59

Dilations, 64Dirac comb, 81Dispersion relation, 51

Fock space, 47, 48Frame, 3Fredholm alternative, 3

Gaussian function, 60Gaussian measure, 47Gram-Schmidt orthogonalization pro-

cess, 11Gram-Schmidt technique, 16

Haar scaling function, 64Hardy space, 6Harper operator, 55Hat function, 58

Hilbert transform, 66

Inverse wavelet transform, 64

Kernel, 3Kernel function, 61

Laguerre polynomials, 43, 85laguerre polynomials, 34Legendre polynomials, 25, 31, 55, 84Legrendre polynomials, 32Lie group, 48

Momentum operator, 72Morlet wavelet, 87Mutually unbiased basis, 19

Null space, 3

Paley-Wiener space, 39Parseval’s relation, 44Peakon, 81Poisson equation, 83Position operator, 42projection theorem, 4

Range, 3Reproducing kernel, 48Reproducing kernel Hilbert space, 39Resolvent, 56

Schauder basis, 37Schwarz inequality, 2Schwarz-Cauchy inequality, 2Shannon wavelet, 67Shifted Legendre polynomials, 28

93

94 Index

Sinc function, 39Sobolev space, 77Spectral theorem, 12Spherical harmonics, 74Symplectic tomogram, 75

Translations, 64Triangle function, 37Triangle inequality, 2

Volterra integral equation, 29

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