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I am planning to study Fourier Analysis from a mathematical point of view. I know that there are some pre-requisites, such as, elements of Functional Analysis and Complex Analysis.

However, I would like to know what exactly I need to be familiar with before studying Fourier Analysis. I will be studying Fourier Analysis from Stein-Shakarchi.

I have done the whole of baby Rudin except Stieltjes Integrals (i.e. the last chapter).

I do not know anything about Complex Analysis or Functional Analysis.

Please tell me what I need to know on these two subjects because I have to make my base strong. I shall be consulting Rudin for Complex Analysis and Conway for Functional Analysis.

Thanks!

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    $\begingroup$ Volume II, Fourier Analysis and Self-Adjointness, of Methods of Modern Mathematical Physics by Reed and Simon might be useful. In their series you can also find the relevant functional analysis. $\endgroup$ – Urgje Nov 17 '15 at 15:05
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    $\begingroup$ Assuming you have some familiarity with Stokes' Theorem, the Divergence Theorem in $\mathbb{R}^3$, I'd say you're well prepared to dive into Volume I right away. Only basic facts of Complex Variables are needed. In fact, the pace may seem slow compared to Rudin (and what wouldn't?) It seems to me that your plan fits well with your background. Stein-Shakarchi is a good choice, but please take a close look at the spelling of the co-author's name. :) $\endgroup$ – DisintegratingByParts Nov 17 '15 at 18:23
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    $\begingroup$ @TrialAndError Thank you for your suggestions. Sure, I am editing the name :) But could you kindly clarify what you meant by "Volume I"? $\endgroup$ – Andrew Richards Nov 17 '15 at 18:32
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    $\begingroup$ The material in Rudin chapters 6-7 is used extensively in Stein and Shakarchi, although they don't use the Riemann-Stieltjes integral, just the plain old Riemann integral. Assuming you are familiar with this material, Stein and Shakarchi should be quite accessible to you. Even a book like Spivak's Calculus would probably be adequate preparation. $\endgroup$ – Bungo Nov 17 '15 at 18:48
  • $\begingroup$ @AndrewRichards : I believe that the Stein-Shakarchi text is Volume I of a 3 volume Princeton Analysis series. $\endgroup$ – DisintegratingByParts Nov 17 '15 at 19:05

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