I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular subject, the better.

Background: For my diploma thesis, I need in particular to understand asymptotics of the Fourier coefficients for certain entire functions, so I need to study it fast, that is, more straightforward, well-structured theory without much "bla-bla", and less exercises... Nevertheless, I would like to learn the more general theory of Fourier analysis for complex/holomorphic functions as it has a great deal of applications in Analytic Number Theory, which is one of the subjects of interest to me.

Thanks in advance!

  • $\begingroup$ PS: How do I make this community wiki? $\endgroup$ – M.G. Mar 8 '11 at 17:36
  • 3
    $\begingroup$ you flag it for moderator attention. And then one of us will ride to your rescue :-) $\endgroup$ – Willie Wong Mar 8 '11 at 17:46
  • $\begingroup$ Related: math.stackexchange.com/questions/4422/… $\endgroup$ – Aryabhata Mar 8 '11 at 17:49
  • $\begingroup$ Just to clarify: I am NOT interested in texts on general/abstract/real Fourier analysis. What I am interested in, is Fourier analysis for complex-valued functions defined on domains in the complex plane, in particular holomorphic functions. @Willie Wong: Thanks for mentioning it! :-) From MO I was kind of used to make my threads CW by myself :-) $\endgroup$ – M.G. Mar 8 '11 at 18:23
  • $\begingroup$ @ex-falso: That is why this question did not get any close votes... I only added that comment, so that you get a convenient link to that question on the Linked section on the right side of this page (that question would get a link to this question too). $\endgroup$ – Aryabhata Mar 8 '11 at 19:04

The following references cover some close links between harmonic and complex analysis that may be suitable for what you need (such as Paley-Wiener theorems, Corona Theorems, etc):

  • Geometric Function Theory: Explorations in Complex Analysis by Steven Krantz

  • Bounded Analytic Functions by John Garnett

  • A Guide to Distribution Theory and Fourier Transforms by Robert Strichartz

  • Real and Complex Analysis by Walter Rudin


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.