As the title indicated, I am wondering if one (probably as an undergraduate math major) can learn much of Fourier Analysis, without taking a course in integration theory.

I am taking a very light introductory course in Hilbert space (using N. Young as the textbook), which does not need any integration theory as pre-requisite. Oftentimes I saw something like "the function behaves like this almost everywhere and hence [some properties]". So please perhaps answer this question:does one really need integration theory for Hilbert space theory, and further functional analysis?

Also for a side note, it gives me the impression that my engineer friends (EE major as such) can deal with Fourier transform/convolution/etc rather comfortably.

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    $\begingroup$ I would recommend against Fourier analysis without measure/integration theory. However, you can learn many qualitative properties of the Fourier transform (enough to use it proficiently and usefully) without having to know anything about measure/integration (perhaps this is what your EE friends are familiar with). $\endgroup$ – parsiad Mar 21 '16 at 19:49
  • $\begingroup$ By "deal", do you just mean that they compute transforms of convenient functions and use them to solve certain differential equations? That's not Fourier analysis; that's running a version of Mathematica with pencil and paper. To answer your question more directly: Yes, you need some familiarity with integration theory to cover the subject properly. $\endgroup$ – anomaly Mar 21 '16 at 19:50
  • $\begingroup$ @anomaly yes indeed that's what I meant by "deal". Would you provide an example or reference to some examples, so that I can see where one really need measure/integration in Fourier analysis/ functional analysis? $\endgroup$ – W.W Mar 21 '16 at 19:53
  • $\begingroup$ Even the answer to the question "for what functions does the defining integral $\hat{f}(u) = \int_{-\infty}^{\infty}f(x) e^{-iux} dx$ make sense?" is the same as the answer to "for what functions does $\int_{-\infty}^{\infty}|f(x)| dx$ exist"? The answer to this depends on which integral you use. You can use the Riemann integral, but then you don't get very good convergence theorems to allow you to interchange limits and integrals, and this is really at the heart of Fourier analysis. So it's much better to study the basics of Lebesgue integration first. $\endgroup$ – Bungo Mar 21 '16 at 19:58
  • $\begingroup$ As in, something beyond just a calculation that treats the Fourier transform as a black box? Sure, see Schwartz's theorem, for example. $\endgroup$ – anomaly Mar 21 '16 at 19:59

It depends on the level you wish to study the fouriertransformation and on how rigorous you want the therory to be. There is a lot of theory and notion on fouriertransformation(and fourierseries, a special case) you can understand and learn with your knowledge. But for deeper coherences and more general stuff you will find the need of understanding some integration and measure theory. If you want to learn something without measuretheory background I recommend books/lectures written for undergraduate physicists, but it won't be that rigorous and more notion. If you wish to dive in deeper using basic knowledge of measure theory I can recommend the first Chapters of "Fourieranalysis on Groups" written by Rudin.

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  • $\begingroup$ Thanks for the reference! This is the only book I've seen written by Rudin except for the baby, big and grandpa. $\endgroup$ – W.W Mar 21 '16 at 21:57

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