- #1

- 290

- 25

- 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!

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!