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Exercice 1

Part 1

Let f : R C 2-periodic, piecewise continuous over [0, 2]. We set

ck(f) :=1

2

2

0

f(t)eiktdt, k Z,

Sn(f)(x) :=n1k=0

ck(f)eikx, Tn(f)(x) :=

1

n

n1k=0

Sk(f)(x), n N, x R.

1) Show that for all x R, for all n N, Sn(f)(x) :=

2

0f(t)Dn(xt)dt, resp. Tn(f)(x) :=

20 f(t)Fn(x t)dt, where the functions Dn and Fn are defined by

Dn(x) :=1

2

n1k=0

eikx, resp. Fn(x) =1

2n

n1k=0

k1j=0

eijx.

2) Show that limn+2

0|Dn(t)|dt = +.

3) Show that Fn 0,2

0Fn(t)dt = 1, and for all ]0, 2[,

2

Fn(t) dt 0 as n +.

4) Deduce from the previous question the following:

a) If f is continuous at x, the sequence (Tn(f)(x))nN converges to f(x) (x R).

b) If f is continuous over R, the sequence (Tn(f))nN converges uniformly to f over R.

c) If ck(f) = 0 for all k Z, then f = 0 outside its discontinuity points.

5) We assume here that f is continuous, and that the sequence (Sn(f))nN converges point-wise to a function g over R. Show that g = f.

6) We assume here that f is C. Show that for all N N

ck(f) = O(|k|N) as |k| +.

7) We assume here that f is C (still 2-periodic). We introduce for all n N the quantity

Rn(f) :=2

n

nj=0

f

j

n

.

Show that for all N NRn(f)

2

0

f(t) dt

= O(|n|N) as |n| +.

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Part 2

For all z C and R > 0, we note D(z, R) := {z C, |z z| < R}. A function F : C, open subset ofC, is said to be analytic over if: for all z0 , there exists a sequence(a0,n)nN C

N and R0 > 0 such that

F(z) =+n=0

a0,n(z z0)n, z D(z0, R0), D(z0, R0) .

We remind that a power series F(z) =

+n=0 anz

n is analytic over its disk of convergence.We admit the following result, known as the principle of isolated zeros: if F : C, open subset of C, is analytic over , either F is zero over , or the set of zeros of F

{z , F(z) = 0} has no accumulation point in .

1) Let R > 0. We say that f : R C est R-analytic if: for all x0 R, there exists a sequence

(a0,n)nN CN such that: f(x) =

+n=0

a0,n (x x0)n, x ]x0 R, x0 + R[.

a) Using the principle of isolated zeros, show that if f is R-analytic, the formula

F(z) :=+n=0

a0,n (z x0)n, z D(x0, R), x0 R

defines an extension of f to the set HR := {z = x + iy, |y| < R}.

b) Show that if f is 2- periodic, F satisfies F(z + 2) = F(z) for all z HR.

2) We define a closed path as the image of an application : [0, 1] C, continuous andpiecewise C1, satisfying (0) = (1). For all F continuous over , we define the integral of Falong as the quantity

F :=

1

0

F((t))(t) dt.

Show that for a power series F(z) =

+n=0 anz

n with radius R > 0, and for all closed path D(0, R), one has

F = 0.

3) We assume here that f : R C is both 2-periodic, and Ranalytic for some R > .Using questions 1)b) and question 2) of part 2, show that there is > 0 such that

ck(f) = O(e|k|), as |k| +.

4) Let f : R C 2-periodic and piecewise continuous. We assume that there exists > 0such that

ck(f) = O(e|k|), lorsque |k| +.

Show that f is R-analytic for some R > 0.

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Exercice 2

In this problem, K will stand for either the field R of real numbers or the field C ofcomplex numbers. Let n 1 be an integer. We let Rn (respectively Cn) be endowed with thestandard euclidean norm (respectively hermitian norm) and Mn(K) be the vector space ofsquare matrices of order n with coefficient in K, endowed with its norm topology. We denoteGLn(K) the group of invertible matrices in Mn(K) and Id GLn(K) the identity matrix.

We will denote det(M) for the determinant of a matrix M, tMfor its transposed and[M, N] := M N N M for the commutator of two matrices M, N. Also the exponentialmatrix of M, denoted exp(M), is the following matrix:

exp(M) :=n=1

Mn

n!.

I) Preliminaries: 1. Show that exp(M) is well defined (that is to say, the series is con-vergent) and that the function M exp(M) is a continuous function Mn(K) Mn(K).

2. Prove that for any M Mn(K), exp(M) GLn(K).

3. Compute det

exp(M)

as a function of the trace T r(M) of M.

4. Prove that, for any M, N, one has

exp

tM)exp(tN) = exp

t(M + N) +t2

2[M, N] + O(t3)

when t R+ goes to 0.

5. Deduce that

limk+

exp

Mk

exp

Nk

exp

Mk

exp

Nk

k2= exp

[M, N]

.

6. Let G be a closed subgroup of GLn(R). Show that

LG := {M Mn(R) / t R, exp(tM) G}

is a sub-vector space of Mn(R).

II) Decompositions of unitary matrices Let U Un(C) = {M Mn(C) / tM M = Id}be unitary. Here, we write M for the matrices obtained by taking the complex conjugateof each entry of M Mn(C).

1. Prove that there exists an orthonormal basis ofCn whose elements are eigenvectorsof U.

2. Let A be in Un(C).

(a) Show that there exists an unitary matrix Q and a diagonal matrix withentries in [0, 1) R, such that A = Q1 exp(2i)Q.

(b) Assume A is both unitary and symmetric. Show that the matrix X = Q1Qis in M

n(R) and is symmetric. Also prove that A = exp(2iX).

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3. Let U Un(C) be unitary.

(a) Prove that the matrix tU U is both unitary and symmetric.

(b) Prove that there exists a symmetric matrix X in Mn(R) and an orthogonalmatrix P On(R) = {M Mn(R) /

tM M = Id} such that U = P exp(iX).

(c) Is such a decomposition unique ?

III) Abelian subgroups of GLn(R) Let G be a connected, closed and commutative sub-group of GLn(R). Recall the vector space LG defined in question I.6).

1. Show that the exponential function M exp(M) is a group homomorphism fromLG (endowed with the addition) to G.

2. Deduced that exp(LG) = G.

3. Show that G = {X LG / exp(X) = Id} is a discrete subgroup of LG, that is

to say a subgroup of LG such that for any x G, the singleton {x} is an opensubset of G (endowed with the subspace topology).

4. Show that there are two integers s, t 0 such that G is isomorphic to the groupRs (S1)t, where S1 is the subgroup ofC {0} of complex numbers of norm 1.

IV) Application to dAlembert-Gauss Theorem Recall that C is a R-vector space ofdimension 2. We let G be the (multiplicative) group C (endowed with the subspacetopology). For any z in G, we denote (z) : C C the function mapping any z in Cto (z)(z) := zz C.

1. Prove that (z) lies in GL2(R), and show that the image (G) of G is a closed

subgroup of GL2(R) which is isomorphic to G.2. Let L R be a commutative field whose dimension as an R-vector space is d 2.

Show that the group G = L is isomrophic to a connected, closed, commutativesubgroup of GLd(R).

3. We recall that for d 3 et t 1, Rd \ {0} is not homeomorphic to Rs (S1)t

(the first space is simply connected while the other is not). Show that d = 2, anddeduce that the field C is algebraically closed (one can use question III.4).

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