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1

$begingroup$

I have a lot of comprehension questions that I can't really figure out by googling:

1. The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference: transforms are for continous cases and series are for discrete cases ?




2. What functions can be Fourier transformed? Which not? I read/learned once that piecewise continuity is enough for a function to be transformed as long as it can be extended to 2 - pi periodic continuity (so the pieces can be "stringed together to a periodic wave")

Is this correct?

functional-analysis fourier-analysis fourier-transform

share|cite|improve this question

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

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  1. is more or less correct but 2. not so much. To be fourier transformed a function has to have some integrability properties because the fourier transform is defined as an integral over all of $mathbbR$.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:08
  

  
  
  
  


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  $begingroup$
  thanks for your prompt response. i'm not looking for complicated counter examples and such, just the most generally applicable case , as in, what is the "weakest" function, that is still transformable
  $endgroup$
  – shakshoka1234
  Mar 21 at 13:22
  

  
  
  
  


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  $begingroup$
  We can readily talk about the fourier transform of $L^1$ functions because $|e^2pi i x xi|leq 1$ and we can use a density argument to extend to $L^2$. Beyond that we can go to distributions and say something about Schwartz functions but I don't really know what is the "weakest' function for this.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So all L1 and L2 functions are fourier transformable?
  $endgroup$
  – shakshoka1234
  Mar 21 at 13:39
  

  
  
  
  


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  $begingroup$
  Yes. The proof is not so bad, you first develop the theory in $L^1$, then look at functions in $L^1cap L^2$, and use a density argument to extend this to $L^2$ itself.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:55
  

  
  
  
  

 | 
show 3 more comments

1

$begingroup$

I have a lot of comprehension questions that I can't really figure out by googling:

1. The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference: transforms are for continous cases and series are for discrete cases ?




2. What functions can be Fourier transformed? Which not? I read/learned once that piecewise continuity is enough for a function to be transformed as long as it can be extended to 2 - pi periodic continuity (so the pieces can be "stringed together to a periodic wave")

Is this correct?

functional-analysis fourier-analysis fourier-transform

share|cite|improve this question

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  1. is more or less correct but 2. not so much. To be fourier transformed a function has to have some integrability properties because the fourier transform is defined as an integral over all of $mathbbR$.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thanks for your prompt response. i'm not looking for complicated counter examples and such, just the most generally applicable case , as in, what is the "weakest" function, that is still transformable
  $endgroup$
  – shakshoka1234
  Mar 21 at 13:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  We can readily talk about the fourier transform of $L^1$ functions because $|e^2pi i x xi|leq 1$ and we can use a density argument to extend to $L^2$. Beyond that we can go to distributions and say something about Schwartz functions but I don't really know what is the "weakest' function for this.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So all L1 and L2 functions are fourier transformable?
  $endgroup$
  – shakshoka1234
  Mar 21 at 13:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes. The proof is not so bad, you first develop the theory in $L^1$, then look at functions in $L^1cap L^2$, and use a density argument to extend this to $L^2$ itself.
  $endgroup$
  – Tony S.F.
  Mar 21 at 13:55
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

I have a lot of comprehension questions that I can't really figure out by googling:

1. The difference between a Fourier transformation and a Fourier series. Am I correct in thinking that the difference: transforms are for continous cases and series are for discrete cases ?




2. What functions can be Fourier transformed? Which not? I read/learned once that piecewise continuity is enough for a function to be transformed as long as it can be extended to 2 - pi periodic continuity (so the pieces can be "stringed together to a periodic wave")

Is this correct?

functional-analysis fourier-analysis fourier-transform

share|cite|improve this question

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

$endgroup$

share|cite|improve this question

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

share|cite|improve this question

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61

asked Mar 21 at 13:06

shakshoka1234shakshoka1234

61