6
$\begingroup$

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!

  • 0
    PS: How do I make this community wiki?2011-03-08
  • 3
    you flag it for moderator attention. And then one of us will ride to your rescue :-)2011-03-08
  • 0
    Related: http://math.stackexchange.com/questions/4422/fourier-analysis-textbook-recommendation2011-03-08
  • 0
    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 :-)2011-03-08
  • 0
    @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).2011-03-08
  • 0
    @Moron: no problem, I just thought some additional clarification would not hurt :-)2011-03-08
  • 0
    For clarification: what exactly do you mean by Fourier analysis for holomorphic functions? Are you talking about taking Fourier transforms over particular slices?2011-03-09

1 Answers 1

1

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