Trigonometric functions and Fourier seriesvipul/studenttalks/introtofourierseries.pdfTrigonometric functions and Fourier series Vipul Naik Periodic functions on reals Homomorphisms

Trigonometricfunctions andFourier series

Vipul Naik

Periodic functionson reals

Homomorphisms

Fourier series incomplex numberslanguage

Quick recap

Rollback to reallanguage

What do we meanby infinite sum?

A little vector spacetheory

Infinite sums in avector space

The inner productspace of periodicfunctions

Definition of an innerproduct

Finding a basis

A basis for the innerproduct

Finding thecoefficients

General theory ininner product spaces

In the case of Fourierseries

Trigonometric functions and Fourier series

Vipul Naik

February 11, 2007


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Outline

Periodic functions on reals

Homomorphisms

Fourier series in complex numbers languageQuick recap

Rollback to real language

What do we mean by infinite sum?A little vector space theoryInfinite sums in a vector space

The inner product space of periodic functionsDefinition of an inner product

Finding a basisA basis for the inner product

Finding the coefficientsGeneral theory in inner product spacesIn the case of Fourier series


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Periodic functions

Let X be a set and f : R → X be a function. Then a numberh ∈ R is termed a period(defined) for f if for any x ∈ R:

f (x + h) = f (x)

A fundamental period(defined) for f is a positive period suchthat there exists no smaller positive period.


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Periodic functions

Let X be a set and f : R → X be a function. Then a numberh ∈ R is termed a period(defined) for f if for any x ∈ R:

f (x + h) = f (x)

A fundamental period(defined) for f is a positive period suchthat there exists no smaller positive period.


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The period group

The set of all periods of a function from R is a group underaddition. That is, the following are true:

I 0 is a period for any function f

I If h is a period for f , so is −h

I If h1 and h2 are periods for f , so is h1 + h2

This is termed the period group(defined) of f .


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The period group







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The period group







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The period group







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The period group







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What kind of subgroups?

Suppose f is a continuous function, and further suppose thath1, h2, . . . , is a sequence of periods of f . Then, for any x :

f (x + h1) = f (x + h2) . . .

hn → h =⇒ x + hn → x + h =⇒ f (x + hn) → f (x + h)

Since f is continuousThis gives:

f (x) = f (x + h)


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f (x + h1) = f (x + h2) . . .


Since f is continuous

This gives:

f (x) = f (x + h)


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f (x + h1) = f (x + h2) . . .


Since f is continuousThis gives:

f (x) = f (x + h)


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Period groups of continuous functions are closed

The upshot of the previous slide is that the limit of anysequence of periods is also period.

Hence, the period group of the continuous function on R isa closed subgroup of R.


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Period groups of continuous functions are closed

The upshot of the previous slide is that the limit of anysequence of periods is also period.Hence, the period group of the continuous function on R isa closed subgroup of R.


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What are the closed subgroups of R?

There are only three possibilities for closed subgroups of R:

I The whole of R

I A subgroup of the form mZ where m ∈ R, viz a discretesubgroup comprising integral multiples of m 6= 0

I The trivial subgroup, that is, the subgroup comprisingonly the zero element


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I The whole of RI A subgroup of the form mZ where m ∈ R, viz a discrete

subgroup comprising integral multiples of m 6= 0



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I The whole of RI A subgroup of the form mZ where m ∈ R, viz a discrete

subgroup comprising integral multiples of m 6= 0



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Classification of continuous functions based onperiod group

A continuous function, based on its period group, can beclassified as:

I A constant function(defined): The period group is RI A periodic function(defined): The period group is mZ for

m > 0. This m is the fundamental period

I A non-periodic function: The period group is trivial


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In the language of groups

Any function on R can also be viewed as a function on thecoset space of its period group. That’s because by its verydefinition, the period group is the group such that thefunction is constant on every coset.

Since R is an Abelian group, the coset space is actually agroup. Thus, the study of periodic continuous functions onR is equivalent to the study of continuous functions on thegroup R/mZ.


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In the language of groups

Any function on R can also be viewed as a function on thecoset space of its period group. That’s because by its verydefinition, the period group is the group such that thefunction is constant on every coset.Since R is an Abelian group, the coset space is actually agroup. Thus, the study of periodic continuous functions onR is equivalent to the study of continuous functions on thegroup R/mZ.


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The coset space is the circle group

Some observations:

I By composing on the right with an appropriate scalarmultiplication, we can normalize the period to somevalue, say 2π.That is, if f has period m, the mapx 7→ f (2πx/m) has period 2π.

I Suppose m = 2π. Then consider the map R to S1 thatsends x ∈ R to (cos x , sin x). Clearly, this map isperiodic with fundamental period 2π. Further, if S1 isviewed as a group of rotations, the map is anisomorphism.

I Thus, the study of periodic continuous functions on Ris the same as the study of continuous functions on S1.


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Some observations:





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Some observations:





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Homomorphisms from R to R

Question: What are the continuous homomorphisms from Rto itself (as a group)?

Since Q is dense in R, any continuous homomorphism fromR to itself is completely determined by its behaviour on Q.Thus, it suffices to determine the possible homomorphismsfrom Q to R.By group-theoretic considerations, any homomorphism fromQ to R is of the form x 7→ λx . Hence, any homomorphismfrom R to R is also of the form x 7→ λx .


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Question: What are the continuous homomorphisms from Rto itself (as a group)?Since Q is dense in R, any continuous homomorphism fromR to itself is completely determined by its behaviour on Q.Thus, it suffices to determine the possible homomorphismsfrom Q to R.

By group-theoretic considerations, any homomorphism fromQ to R is of the form x 7→ λx . Hence, any homomorphismfrom R to R is also of the form x 7→ λx .


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Question: What are the continuous homomorphisms from Rto itself (as a group)?Since Q is dense in R, any continuous homomorphism fromR to itself is completely determined by its behaviour on Q.Thus, it suffices to determine the possible homomorphismsfrom Q to R.By group-theoretic considerations, any homomorphism fromQ to R is of the form x 7→ λx . Hence, any homomorphismfrom R to R is also of the form x 7→ λx .


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Homomorphisms from R to S1

The picture is like this:

R R↓ ↓

S1 S1

The downward maps are the quotient maps.

The question: given a continuous homomorphism from R toS1 (top left to bottom right) can we obtain a continuoushomomorphism from R to R (top left to top right) such thatthe diagram commutes?Answer: YesThus any homomorphism from R to S1 looks likex 7→ (cos λx , sin λx)


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R R↓ ↓

S1 S1

The downward maps are the quotient maps.The question: given a continuous homomorphism from R toS1 (top left to bottom right) can we obtain a continuoushomomorphism from R to R (top left to top right) such thatthe diagram commutes?

Answer: YesThus any homomorphism from R to S1 looks likex 7→ (cos λx , sin λx)


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R R↓ ↓

S1 S1

The downward maps are the quotient maps.The question: given a continuous homomorphism from R toS1 (top left to bottom right) can we obtain a continuoushomomorphism from R to R (top left to top right) such thatthe diagram commutes?Answer: Yes

Thus any homomorphism from R to S1 looks likex 7→ (cos λx , sin λx)


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R R↓ ↓

S1 S1

The downward maps are the quotient maps.The question: given a continuous homomorphism from R toS1 (top left to bottom right) can we obtain a continuoushomomorphism from R to R (top left to top right) such thatthe diagram commutes?Answer: YesThus any homomorphism from R to S1 looks likex 7→ (cos λx , sin λx)


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Homomorphisms from S1 to S1


R R↓ ↓

S1 S1

The downward maps are the quotient maps.

Given a continuous homomorphism φ : S1 → S1, composingφ on the right with the projection from R to S1 gives acontinuous homomorphism from R to S1.This combined with the previous result, tells us that anyhomomorphism from S1 to S1 is of the form:

(cos x , sin x) 7→ (cos nx , sin nx)


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R R↓ ↓

S1 S1

The downward maps are the quotient maps.Given a continuous homomorphism φ : S1 → S1, composingφ on the right with the projection from R to S1 gives acontinuous homomorphism from R to S1.

This combined with the previous result, tells us that anyhomomorphism from S1 to S1 is of the form:



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R R↓ ↓

S1 S1

The downward maps are the quotient maps.Given a continuous homomorphism φ : S1 → S1, composingφ on the right with the projection from R to S1 gives acontinuous homomorphism from R to S1.This combined with the previous result, tells us that anyhomomorphism from S1 to S1 is of the form:



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In terms of complex numbers

The plane R2 can be viewed as the complex numbers C,that is, every point (x , y) can be identified with the complexnumber x + iy . Under this identification, S1 is a subgroup ofthe multiplicative group of nonzero complex numbers.

In this language, then (cos x , sin x) is the same as ex . Thus,homomorphisms from S1 to S1 are maps of the form:

ex 7→ enx

which is the same as:

z 7→ zn


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In terms of complex numbers

The plane R2 can be viewed as the complex numbers C,that is, every point (x , y) can be identified with the complexnumber x + iy . Under this identification, S1 is a subgroup ofthe multiplicative group of nonzero complex numbers.In this language, then (cos x , sin x) is the same as ex . Thus,homomorphisms from S1 to S1 are maps of the form:

ex 7→ enx

which is the same as:

z 7→ zn


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The general setup

1. We wanted to understand continuous periodic functionsfrom R to X .

2. We converted this to understanding continuousfunctions from R/2πZ to X .

3. We converted this to understanding continuousfunctions from S1 to X .


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The general setup





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The general setup





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What we’re interested in

The situation that we want to study is where X = C, viz theproblem of continuous periodic functions from R to C. Thisreduces to the problem of all continuous functions from S1

to C.

We already have a bunch of continuous functions from S1 toS1 (which is a subset of C), namely: the maps z 7→ zn.These maps are called characters.


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What we’re interested in

The situation that we want to study is where X = C, viz theproblem of continuous periodic functions from R to C. Thisreduces to the problem of all continuous functions from S1

to C.We already have a bunch of continuous functions from S1 toS1 (which is a subset of C), namely: the maps z 7→ zn.These maps are called characters.


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Are all continuous functions expressible viacharacters?

Which continuous maps from S1 to C can be expressed interms of characters?

I Polynomial maps are expressible as finite linearcombinations of characters

I Laurent polynomial maps are expressible as finite linearcombinations of characters

I Maps which have expressions as power series or asLaurent series about the origin, can be expressed asinfinite linear combinations of characters

Do all continuous maps fall in the third class? Can everycontinuous map on S1 be expressed as a (possibly infinite)sum of characters?


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Continuous functions to reals

Let’s now consider the case where X = R, that is, theproblem of determining continuous functions from S1 to R.Clearly, we cannot directly use characters since charactersare maps from S1 to C. However, we can project thecharacters on the axes, viz take their coordinates, and obtainthe following collections of periodic functions:

z 7→ Re zn and z 7→ Im zn

for n ∈ ZBy thinking of these as periodic functions on R, we get thebunches:

x 7→ cos nx and x 7→ sin nx


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for n ∈ Z

By thinking of these as periodic functions on R, we get thebunches:



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The question for reals

The question for reals is: given a periodic function R → R,what are the conditions under which it can be expressed asan infinite linear combination of the cosine and sine bunches?


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What is a vector space?

Let k be a field (such as R, C). Then a vector space over k(also, a k-vector space) is a set V equipped with:

I An additive operation + under which V is an Abeliangroup

I A scalar multiplication action of k on V with theproperty that the scalar multiplication by any elementinduces an Abelian group homomorphism on V

We can thus talk of finite k-linear combinations on V .There is, however, no inherent meaning associated to infinitek-linear combinations on V .


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Linear independence and basis

A subset S of a vector space V is said to be linearlyindependent if 0 cannot be expressed as a nontrivial k-linearcombination of any finite subset of S , that is, if:

r∑i=1

aivi = 0

for vi ∈ S and ai ∈ k, then each ai = 0The span(defined) of a subset S is defined as the vectorsubspace of V containing all those elements that are finitek-linear combinations of elements of S .A basis(defined) of a vector space is a linearly independentsubset whose span is the whole vector space.


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Inner products and orthogonal vectors

An inner product(defined) on a real vector space is ageneralization of the dot product that we’ve usually seen.(definition later).

Two vectors are said to be orthogonal(defined) if their innerproduct is zero.Now for an observation: any family of elements that ispairwise orthogonal with respect to an inner product, is alsolinearly independent.


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An inner product(defined) on a real vector space is ageneralization of the dot product that we’ve usually seen.(definition later).Two vectors are said to be orthogonal(defined) if their innerproduct is zero.

Now for an observation: any family of elements that ispairwise orthogonal with respect to an inner product, is alsolinearly independent.


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An inner product(defined) on a real vector space is ageneralization of the dot product that we’ve usually seen.(definition later).Two vectors are said to be orthogonal(defined) if their innerproduct is zero.Now for an observation: any family of elements that ispairwise orthogonal with respect to an inner product, is alsolinearly independent.


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Orthonormal basis

An orthonormal basis is a basis where the elements arepairwise orthogonal.

Given a vector space and an inner product, can we find anorthonormal basis for the vector space?The answer is always yes for a finite-dimensional vectorspace. The standard technique is Gram-Schmidtorthogonalization.


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Orthonormal basis

An orthonormal basis is a basis where the elements arepairwise orthogonal.Given a vector space and an inner product, can we find anorthonormal basis for the vector space?

The answer is always yes for a finite-dimensional vectorspace. The standard technique is Gram-Schmidtorthogonalization.


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Orthonormal basis

An orthonormal basis is a basis where the elements arepairwise orthogonal.Given a vector space and an inner product, can we find anorthonormal basis for the vector space?The answer is always yes for a finite-dimensional vectorspace. The standard technique is Gram-Schmidtorthogonalization.


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Topological vector space

A topological vector space is a vector space with a topologygiven both to the vector space and to the base field suchthat:

I All the field operations are continuous with respect tothe topology given to the base field.

I The group operations on the vector space arecontinuous with respect to the topology on the vectorspace.

I The scalar multiplication operation is continuous fromthe product of the field and the vector space to thevector space.


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Infinite sums in a topological vector space

Given a sequence of vectors v1, v2, and so on in atopological vector space, the infinite sum

∑∞i=1 vi is defined

as the limit of the partial sums:

∞∑i=1

vi = limn→∞

n∑i=1

vi

Note that the infinite sum makes no sense without thetopology because it depends on the notion of limit.


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∞∑i=1

vi = limn→∞

n∑i=1

vi



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∞∑i=1

vi = limn→∞

n∑i=1

vi



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Normed linear space

Consider an R-vector space V . A norm function on thisvector space associates to each v ∈ V a nonnegative realnumber N(v) such that:

I N(v) = 0 =⇒ v = 0

I N(λv) = |λ|N(v)

I The map (v ,w) 7→ N(v − w) defines a metric on V .Equivalently, for any vectors v ,w ∈ V :

N(v + w) ≤ N(v) + N(w)


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Normed linear space


I N(v) = 0 =⇒ v = 0

I N(λv) = |λ|N(v)


N(v + w) ≤ N(v) + N(w)


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Normed linear space


I N(v) = 0 =⇒ v = 0

I N(λv) = |λ|N(v)


N(v + w) ≤ N(v) + N(w)


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The Lr -normed linear spaces

Examples of norms are the Lr -norms, in finite-dimensionalvector spaces. Take a basis e1, e2, . . . , en of V and for anyv ∈ V , consider the unique expression:

v =n∑

i=1

lambdaiei

Now define:

N(v) :=

(n∑

i=1

λri

)1/r

For r ≥ 2, N is a norm.


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The Lr -normed linear spaces

Examples of norms are the Lr -norms, in finite-dimensionalvector spaces. Take a basis e1, e2, . . . , en of V and for anyv ∈ V , consider the unique expression:

v =n∑

i=1

lambdaiei

Now define:

N(v) :=

(n∑

i=1

λri

)1/r

For r ≥ 2, N is a norm.


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Topology from the norm

Any normed R-vector space automatically gets the structureof a topological vector space, because, first of all, it gets thestructure of a metric space, and any metric space naturallycomes with a topology.

Thus, we can talk of notions of infinite sums andconvergence for normed vector spaces.


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Topology from the norm

Any normed R-vector space automatically gets the structureof a topological vector space, because, first of all, it gets thestructure of a metric space, and any metric space naturallycomes with a topology.Thus, we can talk of notions of infinite sums andconvergence for normed vector spaces.


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Inner product spaceLet V be a R-vector space. Then an inner product on Vassociates to every pair of vectors v ,w ∈ V a real number〈 v , w 〉 such that:

I It is linear in the first variable:

〈 v1 , w 〉+ 〈 v2 , w 〉 = 〈 v1 + v2 , w 〉

and〈 λv , w 〉 = λ 〈 v , w 〉

I It is symmetric, viz:

〈 v , w 〉 = 〈 w , v 〉

I It is positive definite, viz:

〈 v , v 〉 > 0

for v 6= 0


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〈 v1 , w 〉+ 〈 v2 , w 〉 = 〈 v1 + v2 , w 〉

and〈 λv , w 〉 = λ 〈 v , w 〉


〈 v , w 〉 = 〈 w , v 〉


〈 v , v 〉 > 0

for v 6= 0


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〈 v1 , w 〉+ 〈 v2 , w 〉 = 〈 v1 + v2 , w 〉

and〈 λv , w 〉 = λ 〈 v , w 〉


〈 v , w 〉 = 〈 w , v 〉


〈 v , v 〉 > 0

for v 6= 0


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Inner product spaces are normed

Given a space V with an inner product, we can naturallydefine a norm on V by setting N(v) =

√〈 v , v 〉.

Thus, every inner product space is a normed space andhence also a topological vector space.


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Inner product spaces are normed

Given a space V with an inner product, we can naturallydefine a norm on V by setting N(v) =

√〈 v , v 〉.

Thus, every inner product space is a normed space andhence also a topological vector space.


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Concerns when dealing with infinite sums

Having seen a bit of infinite sums on the real line, we shouldbe aware of the different notions of convergence:

1. absolute convergence(defined): Here the series of normsof the vectors is convergent.

2. unconditional convergence(defined): Here anyrearrangement of the vectors gives a convergent series.

3. conditional convergence(defined): Here, the given seriesis convergent, but nothing is guaranteed aboutrearrangements of it.


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The use of Cauchy sequences

A Cauchy sequence in a topological vector space is asequence where for any bound ε, there exists an N such thatfor all m, n > N, the tail sum Sm − Sn is bounded inmagnitude by ε.

Any convergent sequence is Cauchy. If, in a normed vectorspace, the converse is true (viz any Cauchy sequence isconvergent) then the normed vector space is said to becomplete(defined). For instance, any finite-dimensional vectorspace over R is complete.


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The use of Cauchy sequences

A Cauchy sequence in a topological vector space is asequence where for any bound ε, there exists an N such thatfor all m, n > N, the tail sum Sm − Sn is bounded inmagnitude by ε.Any convergent sequence is Cauchy. If, in a normed vectorspace, the converse is true (viz any Cauchy sequence isconvergent) then the normed vector space is said to becomplete(defined). For instance, any finite-dimensional vectorspace over R is complete.


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Inner product defined by an integral

Consider the following integral for functions f , g : S1 → R:∫S1

f (x)g(x) dx

where the measure of integration is the usual arc-lengthfunction on S1.

Alternatively, via the identification of S1

with the interval from 0 to 2π, we have:∫ 2π

0f (x)g(x) dx


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Inner product defined by an integral

Consider the following integral for functions f , g : S1 → R:∫S1

f (x)g(x) dx

where the measure of integration is the usual arc-lengthfunction on S1. Alternatively, via the identification of S1

with the interval from 0 to 2π, we have:∫ 2π

0f (x)g(x) dx


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Some observations about this integral

1. When f and g are continuous functions, the integral iswell-defined.

2. The integral is symmetric and bilinear. Moreover, for acontinuous function f , the value of the integral wheng = f is 0 if and only if f is identically zero. Otherwiseit is positive

Thus, this integral defines an inner product if we restrict tothe subspace defined by continuous functions on S1.


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Notion of convergence for functionsBefore proceeding to study convergence questions, we mustdefine a suitable notion of convergence for functions. Thereare the following notions possible:

I Pointwise convergence: This means that the value ofthe function in the sequence must, at each point,converge to the value of the function.

I Uniform convergence: This means that for any ε, thereis a uniform δ we can choose such that the ε-δcondition holds at every point.

I Lr -norm convergence: This convergence arises from thenaturally defined Lr -norm. The Lr -norm in this case isan infinite-dimensional analogue of the Lr -norm forfinite-dimensional spaces:

f 7→(∫ 2π

0f (x)r dx

)1/r


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f 7→(∫ 2π

0f (x)r dx

)1/r


Vipul Naik


Homomorphisms


Quick recap







Finding a basis









f 7→(∫ 2π

0f (x)r dx

)1/r


Vipul Naik


Homomorphisms


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Uniform convergence and the sup-norm

The supremum of a continuous function from S1 to R isdefined as the maximum of absolute values of elements in itsimage set. The norm that associates to each continuousfunction its supremum, gives rise to the topology of uniformconvergence.

For obvious reasons, the sup-norm is also often called theL∞-norm.We can further show that under uniform convergence, thelimit of any sequence of continuous functions is alsocontinuous. Thus, with respect to the topology of uniformconvergence, the continuous functions form a closedsubspace.


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The supremum of a continuous function from S1 to R isdefined as the maximum of absolute values of elements in itsimage set. The norm that associates to each continuousfunction its supremum, gives rise to the topology of uniformconvergence.For obvious reasons, the sup-norm is also often called theL∞-norm.

We can further show that under uniform convergence, thelimit of any sequence of continuous functions is alsocontinuous. Thus, with respect to the topology of uniformconvergence, the continuous functions form a closedsubspace.


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Homomorphisms


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The supremum of a continuous function from S1 to R isdefined as the maximum of absolute values of elements in itsimage set. The norm that associates to each continuousfunction its supremum, gives rise to the topology of uniformconvergence.For obvious reasons, the sup-norm is also often called theL∞-norm.We can further show that under uniform convergence, thelimit of any sequence of continuous functions is alsocontinuous. Thus, with respect to the topology of uniformconvergence, the continuous functions form a closedsubspace.


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The inner product and the topology induced

The inner product that we considered:∫ 2π

0f (x)g(x) dx

gives rise to the norm:

f 7→

√∫ 2π

0f (x)2 dx

Which is the L2-topology. Thus, this inner product inducesthe topology of L2-convergence.


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Relation between these topologies

It turns out that for r < s, Ls -convergence is a strongercondition than Lr -convergence. This is because forLs -convergence, the sequence must converge rapidly to itslimits even at points where the deviation is much greater.

In particular, uniform convergence (Which isL∞-convergence) is a much stronger condition thanL2-convergence.Thus, while results on L2-convergence come for free whendealing with this inner product, results on uniformconvergence require additional machinery.


Vipul Naik


Homomorphisms


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It turns out that for r < s, Ls -convergence is a strongercondition than Lr -convergence. This is because forLs -convergence, the sequence must converge rapidly to itslimits even at points where the deviation is much greater.In particular, uniform convergence (Which isL∞-convergence) is a much stronger condition thanL2-convergence.

Thus, while results on L2-convergence come for free whendealing with this inner product, results on uniformconvergence require additional machinery.


Vipul Naik


Homomorphisms


Quick recap







Finding a basis






It turns out that for r < s, Ls -convergence is a strongercondition than Lr -convergence. This is because forLs -convergence, the sequence must converge rapidly to itslimits even at points where the deviation is much greater.In particular, uniform convergence (Which isL∞-convergence) is a much stronger condition thanL2-convergence.Thus, while results on L2-convergence come for free whendealing with this inner product, results on uniformconvergence require additional machinery.


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Homomorphisms


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Outline


Homomorphisms








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The cosine and sine functions – orthogonal

The following can be checked:

I ∫ 2π

0cos mx cos nx dx = πδmn

I ∫ 2π

0cos mx sin nx dx = 0

I ∫ 2π

0sin mx sin nx dx = πδmn

Thus, the functions x 7→ cos mx and x 7→ sin nx are pairwiseorthogonal.


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Homomorphisms


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I ∫ 2π


I ∫ 2π


I ∫ 2π




Vipul Naik


Homomorphisms


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I ∫ 2π


I ∫ 2π


I ∫ 2π




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What are the finite linear combinations of these?

A function f : S1 → R that can be expressed as a finitelinear combinations of cos and sin functions is termed atrigonometric polynomial. Given the angle sum formulae, afunction is a trigonometric polynomial if and only if it can beexpressed as a polynomial in the cos and sin functions.Clearly, then, the finite linear combinations are very few.


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Meaning of Fourier series

A Fourier series is an expression as an infinite linearcombination of the sine and cosine terms.

A typical Fourier series:

a0 +∞∑

n=1

(an cos nx + bn sin nx)

Thus, the question of determining which functions can beexpressed as infinite linear combinations reduces to thequestion of determining which functions have Fourier seriesthat converge to them.


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Meaning of Fourier series

A Fourier series is an expression as an infinite linearcombination of the sine and cosine terms.A typical Fourier series:

a0 +∞∑

n=1


Thus, the question of determining which functions can beexpressed as infinite linear combinations reduces to thequestion of determining which functions have Fourier seriesthat converge to them.


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Various kinds of convergence

There are two parts to the question: given a function, canwe associate a unique Fourier series to it? If we are able todo that, we can ask:

I For which functions does the Fourier series converge inthe L2-norm?

I For which functions does the Fourier series convergepointwise?

I For which functions does the Fourier series convergeuniformly?


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Vipul Naik


Homomorphisms


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Vipul Naik


Homomorphisms


Quick recap







Finding a basis





Outline


Homomorphisms








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Homomorphisms


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Expressing a vector using an orthonormal basis

Let v1, v2, . . . , vn form an orthonormal basis for a vectorspace V with respect to an inner product 〈 , 〉. Then, if

v =∑

i

λivi

We have the following formula for λi :

λi = 〈 v , vi 〉

In other words, λi is the length of the projection of v on thei th coordinate.


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Homomorphisms


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Expressing a vector using an orthonormal basis

Let v1, v2, . . . , vn form an orthonormal basis for a vectorspace V with respect to an inner product 〈 , 〉. Then, if

v =∑

i

λivi

We have the following formula for λi :

λi = 〈 v , vi 〉

In other words, λi is the length of the projection of v on thei th coordinate.


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The same for an infinite orthonormal set

Suppose we have an infinite orthonormal set vi indexed byi ∈ I in a topological vector space V . Then, given a vectorv ∈ V , defines λi = 〈 v , vi 〉 as i ∈ I .

Question: Under what conditions is it true that:

v =∑i∈I

λivi


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The same for an infinite orthonormal set

Suppose we have an infinite orthonormal set vi indexed byi ∈ I in a topological vector space V . Then, given a vectorv ∈ V , defines λi = 〈 v , vi 〉 as i ∈ I .Question: Under what conditions is it true that:

v =∑i∈I

λivi


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Linearly dense orthonormal set

For an infinite-dimensional topological vector space, we maynot be able to find an orthonormal basis. However, we maybe able to find an orthonormal set with the property that theinfinite linear combinations of elements in that set cover thewhole space.

This happens if and only if the linear subspace comprisingfinite linear combinations is a dense subspace of the wholespace.The “abstract nonsense” guarantees that the series obtainedin this way converges in the topology induced by the innerproduct.


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Homomorphisms


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For an infinite-dimensional topological vector space, we maynot be able to find an orthonormal basis. However, we maybe able to find an orthonormal set with the property that theinfinite linear combinations of elements in that set cover thewhole space.This happens if and only if the linear subspace comprisingfinite linear combinations is a dense subspace of the wholespace.

The “abstract nonsense” guarantees that the series obtainedin this way converges in the topology induced by the innerproduct.


Vipul Naik


Homomorphisms


Quick recap







Finding a basis






For an infinite-dimensional topological vector space, we maynot be able to find an orthonormal basis. However, we maybe able to find an orthonormal set with the property that theinfinite linear combinations of elements in that set cover thewhole space.This happens if and only if the linear subspace comprisingfinite linear combinations is a dense subspace of the wholespace.The “abstract nonsense” guarantees that the series obtainedin this way converges in the topology induced by the innerproduct.


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Homomorphisms


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Fourier coefficients and L2-convergenceThe Fourier series for a 2π-periodic function f looks like:

a0 +∞∑

n=1


Where we have:

an =1

π

∫ 2π

0f (x) cos(nx) dx

and

bn =1

π

∫ 2π

0f (x) sin(nx)

For any L2-function, the Fourier coefficients are well-definedand the Fourier series converges in the L2-sense.In particular, for continuous functions the Fourier seriesconverges in the L2 sense.


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Homomorphisms


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a0 +∞∑

n=1


Where we have:

an =1

π

∫ 2π

0f (x) cos(nx) dx

and

bn =1

π

∫ 2π

0f (x) sin(nx)



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Homomorphisms


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a0 +∞∑

n=1


Where we have:

an =1

π

∫ 2π

0f (x) cos(nx) dx

and

bn =1

π

∫ 2π

0f (x) sin(nx)

For any L2-function, the Fourier coefficients are well-definedand the Fourier series converges in the L2-sense.

In particular, for continuous functions the Fourier seriesconverges in the L2 sense.


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Homomorphisms


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a0 +∞∑

n=1


Where we have:

an =1

π

∫ 2π

0f (x) cos(nx) dx

and

bn =1

π

∫ 2π

0f (x) sin(nx)



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Homomorphisms


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Conditions for uniform and pointwise convergence

It turns out that for a C 1 function, the Fourier series not onlyconverges to it in the L2 sense, it also converges uniformly.

Further, for a continuous function, it converges pointwise.We shall see proofs of these facts next time.


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Homomorphisms


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It turns out that for a C 1 function, the Fourier series not onlyconverges to it in the L2 sense, it also converges uniformly.Further, for a continuous function, it converges pointwise.

We shall see proofs of these facts next time.


Vipul Naik


Homomorphisms


Quick recap







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It turns out that for a C 1 function, the Fourier series not onlyconverges to it in the L2 sense, it also converges uniformly.Further, for a continuous function, it converges pointwise.We shall see proofs of these facts next time.

Documents

Trigonometric functions and Fourier seriesvipul/studenttalks/introtofourierseries.pdfTrigonometric functions and Fourier series Vipul Naik Periodic functions on reals Homomorphisms