What's the definition of "periodic extension of a function"?

[CGandC]

TL;DR Summary

x

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to:

1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ):
for $f: [ a,b) \to \mathbb{R}$ its periodic extension is defined as $\tilde{f}(x+n(b-a))=f(x) \quad ~~~ , \forall x \in[a, b), \quad n \in \mathbb{Z}$.

2. Why is it necessary to periodically extend a function? In my lecture notes, before calculating $\hat{f}(n)$ ( the Fourier coefficients of a periodic function $f: [ a,b) \to \mathbb{R}$ with period $T$ ) it is said that $f$ should be periodically extended. But I still don't fully understand why a periodic extension is necessary.

3. Is Fourier series considered a periodic extension of a function?. I mean, is the following true?
Suppose $H : \mathbb{R} \to \mathbb{R}$ is given. Also, suppose $f: [ a,b) \to \mathbb{R}$ is given.
$H$ is Fourier series of $f$ $\iff$ $H$ is a periodic extension of $f$

4. Does a function $f$ must have a period in order to be periodically extendible?, according to my definition in question 1 above, the answer's no ( because the period can be defined as the length of the interval ), but still.

Thanks in advance for the help!

 

[fresh_42]

The goal of Fourier transformations is to approximate a function by (periodic) trigonometric functions. So either you have to make your function periodic or restrict the trig functions to a certain interval. The first case seems to be easier and reasonable since we do not have to "redefine" trig functions and can convert them into the exponential function at any time.

 

[CGandC]

So I'm defining a sequence of functions $\{ f_n \}_{ n \in \mathbb{Z} }$ where for every $n \in \mathbb{Z}$, $f_n : [ a+n,b+n) \to \mathbb{R}$, and this sequence is said to be the periodic extension of $f$? or is it that each such function is defined to be a perioidic extension of $f$ ?

Or is it that I'm defining an entire function $F : \mathbb{R} \to \mathbb{R}$ which is defined as the periodic extension of $f$?

I think it's the latter since I need a function defined on $\mathbb{ R}$ so It can be approximated by a span of trigonometric polynomials ( which is then said to be the Fourier series of $f$ ).

 

[fresh_42]

I am not certain and would check it with the books. But I think we simply expand the domain of $f$ by repeating it: $F\, : \,\mathbb{R}\longrightarrow \mathbb{R}$ defined as $F(x)=f(x-kb+ka)\text{ if } x\in [kb-(k-1)a,(k+1)b-ka).$

We won't get continuity at the interval limits, but that shouldn't be a problem.

 

[FactChecker]

I think it's the latter since I need a function defined on $\mathbb{ R}$ so It can be approximated by a span of trigonometric polynomials ( which is then said to be the Fourier series of $f$ ).

The Fourier series will approximate the extended function. You can use it to approximate the original function in the original range, but you should realize that the approximation is actually for the extended function. That explains why the approximation at each endpoint is heavily affected by the values at the other endpoint. If you want a better approximation for the original function in the original range, there are many that will go exactly through the endpoints. But they will not be as useful in analyzing the frequency content of the function.

 

[otennert]

Summary:: x

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to:

1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ):
for $f: [ a,b) \to \mathbb{R}$ its periodic extension is defined as $\tilde{f}(x+n(b-a))=f(x) \quad ~~~ , \forall x \in[a, b), \quad n \in \mathbb{Z}$.

2. Why is it necessary to periodically extend a function? In my lecture notes, before calculating $\hat{f}(n)$ ( the Fourier coefficients of a periodic function $f: [ a,b) \to \mathbb{R}$ with period $T$ ) it is said that $f$ should be periodically extended. But I still don't fully understand why a periodic extension is necessary.

3. Is Fourier series considered a periodic extension of a function?. I mean, is the following true?
Suppose $H : \mathbb{R} \to \mathbb{R}$ is given. Also, suppose $f: [ a,b) \to \mathbb{R}$ is given.
$H$ is Fourier series of $f$ $\iff$ $H$ is a periodic extension of $f$

4. Does a function $f$ must have a period in order to be periodically extendible?, according to my definition in question 1 above, the answer's no ( because the period can be defined as the length of the interval ), but still.

Thanks in advance for the help!

Ad 1: correct. A function that is continuous in a given interval $[a,b)$ is just copied and pasted infinite times so to speak towards $\pm\infty$.

Ad 2: A periodic extension of a function is not necessary, but first of all just a concept. A better question would be: when is a periodic extension an interesting concept? An example where a periodic extension is applied is when the remainder term in the Euler--Maclaurin series is to be given in an explicit form: https://en.wikipedia.org/wiki/Euler–Maclaurin_formula

Ad 3: No. If a function H is given for all $\mathbb{R}$, a periodic extension is simply speaking the restriction of that function on an interval $[a,b)$ and above-mentioned copy & paste patching.

Ad 4: No. The function only needs to be continuous in some half-open interval, which then defines the patch to be copied & pasted.

 

[CGandC]

Ad 1: correct. A function that is continuous in a given interval $[a,b)$ is just copied and pasted infinite times so to speak towards $\pm\infty$.

Ad 2: A periodic extension of a function is not necessary, but first of all just a concept. A better question would be: when is a periodic extension an interesting concept? An example where a periodic extension is applied is when the remainder term in the Euler--Maclaurin series is to be given in an explicit form: https://en.wikipedia.org/wiki/Euler–Maclaurin_formula

Ad 3: No. If a function H is given for all $\mathbb{R}$, a periodic extension is simply speaking the restriction of that function on an interval $[a,b)$ and above-mentioned copy & paste patching.

Ad 4: No. The function only needs to be continuous in some half-open interval, which then defines the patch to be copied & pasted.

Helpful answer, thank you!

 

[FactChecker]

Ad 4: No. The function only needs to be continuous in some half-open interval, which then defines the patch to be copied & pasted.

It does not need to be continuous inside the interval where it is defined.

 

[WWGD]

It does not need to be continuous inside the interval where it is defined.

I think it's usually expected to be in $L^2[a,b)$, or whatever interval it's defined in, right? Maybe in $L^p[a,b)$ for some p? Edit: IIRC, the set $\{ Cos(nx), Sin(nx) n=1,2,...\}$ is dense in $L^2[a,b]$. Sorry for being lazy; it seems you have to filter a lot in web sources to get to this and similar.

 

[FactChecker]

the set $\{ Cos(nx), Sin(nx) n=1,2,...\}$ is dense in $L^2[a,b]$.

I think you mean the set generated by that basis.

 

[wrobel]

Let $f\in L^2(0,2\pi)$. Then the series $\sum_{k\in\mathbb{Z}}f_ke^{ikx}$ converges to a function $g\in L^2_{loc}(\mathbb{R})$ in $L^2_{loc}(\mathbb{R})$. Here $f_k$ are the Fourier coefficients of $f$.
The function $g$ has the following properties:
1) $g(x)=f(x)$ almost everywhere in $(0,2\pi)$ and
2) $g(x+2\pi)=g(x)$ for almost all $x\in\mathbb{R}$.

($L^p(0,2\pi)\subset L^2(0,2\pi)$ for $p\in[2,\infty]$)

Convergence in the space of continuous functions or in $L^p(0,2\pi),\quad p\ne 2$ is a much more tricky story

 

[WWGD]

Yes, sorry, the span of $aCos(nx)+ bSin(nx)$ is dense in $L^2[a,b]$. Yes, Hilbert spaces are just-about always nicer to work with. There is an algorithm to construct approximations in $C[a,b]$ using the dense subset $\{1, x, x^2, ..., x^n,... \}$. Using convolutions, I think.

 

[WWGD]

Let $f\in L^2(0,2\pi)$. Then the series $\sum_{k\in\mathbb{Z}}f_ke^{ikx}$ converges to a function $g\in L^2_{loc}(\mathbb{R})$ in $L^2_{loc}(\mathbb{R})$. Here $f_k$ are the Fourier coefficients of $f$.
The function $g$ has the following properties:
1) $g(x)=f(x)$ almost everywhere in $(0,2\pi)$ and
2) $g(x+2\pi)=g(x)$ for almost all $x\in\mathbb{R}$.

($L^p(0,2\pi)\subset L^2(0,2\pi)$ for $p\in[2,\infty]$)

Convergence in the space of continuous functions or in $L^p(0,2\pi),\quad p\ne 2$ is a much more tricky story

I thought inclusion applied only to bounded intervals. By Holder's theorem, I think.

 

[wrobel]

I thought inclusion applied only to bounded intervals. By Holder's theorem, I think.

yes

There is an algorithm to construct approximations in C[a,b] using the dense subset {1,x,x2,...,xn,...}. Using convolutions, I think.

the set of polynomials is dense , yes

 

[wrobel]

The set of trigonometric polynomials is dense in C[a,b] but not each function from C[a,b] can be expanded into the Fourier series

Gelbaum Olmsted Counterexamples in analysis