Assuming a periodic function, Dirichlet conditions are sufficient (not necessary) conditions for Fourier series.

1) As they are defined for Fourier series, how Dirichlet conditions can also be considered as sufficient conditions for the existence of the Fourier transform (the convergence of the integral)?

2) Do Dirichlet conditions valid only for real-valued functions?

3) Dirichlet conditions include $L^1$ function condition. Is $L^2$ function a sufficient condition for the existence of the Fourier transform?


1 Answer 1


2) No 3) These two are completely different- Dirichlet's condition talks about pointwise convergence while the L2 condition talks about convergence in norm. And neither type of convergence implies the other.

As a senior in high school I sometimes have to interpret a paragraph as a whole in an English test since I might have to do it that way on the NCAT(National College Aptitude Test) on 14 Nov. And even then I might not be able to understand every word (and instead find the meaning out impromptu) When I was in grade school though English classes and tests were mainly about learning English word by word. But knowing the meaning of every word of a paragraph did not guarantee the knowledge of the paragraph.

Interpreting paragraphs in whole = convergence in norm Learning word by word = pointwise convergence; now you can see why neither implies the other.

P.S. no they don't teach that there is any other notion of convergence than the pointwise one at HS level at school even in Korea

But I found out by myself about what a "convergence in norm" is via the Internet: and just typing "Fourier transform" on Google leads you to links to some American colleges' websites (that I don't think at all to apply for lol) on which the L2 Fourier transform is mentioned in.


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