Why Fourier Analysis has so many branches?

Question

I was exposed to Fourier Analysis at my freshmen period when I only knew a little about analysis in one variable. And at my second year, Fourier Analysis appeared in complex analysis and real analysis, and I get to know that one can conduct Fourier Analysis over a topological group. Now, in my third year in college, I think it is not a too late time to learn Fourier Analysis seriously. But I am confused

• Why Fourier Analysis is so useful and so general that it has so many branches.

And I have a look on some of the text in Fourier Analysis, but different books requires different kind of knowledge. Some of them ask for only Mathematics Analysis, and some of them require a good command on Functional Analysis. So I want your recommendation about

• What is the suitable level for high grade undergraduate or graduate students to learn Fourier Analysis seriously?

And clearly, there are numbers of texts for Fourier analysis (at least, they are titled with `Fourier Analysis'). I wonder that

• Which book is a good beginning for high grade undergraduate or graduate level? And are there any book treating different branches properly.

Thank you.

Answer 1

I bet that there are questions on MSE with a very similar topic but I can't find any so..

I would say a good command of one variable-Calculus is the best "lower bound" to begin delving into Fourier Analysis. Having taken a course in Real Analysis could help but it is not necessary. If you are familiar with Measure Theory and in particular Lebesgue Measure-that would be a boon.In any case, third year is certainly not a late time to begin.

Why Fourier transform is so useful?-Take a look at this MSE answer and in particular its last paragraph- Fourier transform for dummies

Considering book recommendations have a look at this one as an introduction-

"Fourier Series" by Georgi Tolstov

And at a more advanced level-

"An Introduction to Harmonic Analysis"

Both can be relatively easily found online.

Answer 2

Fourier analysis is at the very heart of the study of linear systems. These enjoy the superposition principle (the effect of a sum of causes is the sum of the individual effects), so that the decomposition of a signal as a sum of basis function is a natural approach.

Important basis functions are those that satisfy linear differential equations, i.e. complex exponentials, which are free reponses of those systems.

This establishes a bridge to the Laplace transform (decomposition as a sum of complex exponentials, corresponding to the roots of the characteristic polynomial), and its little sister, the Fourier transform. (In the case of an analytical transform, it is enough to consider its values along the imaginary axis.)

Note that when the signal is periodic, the Fourier transform becomes the discrete Fourier Series. And when the signal is discrete, the Fourier transform is... periodic.

In numerical applications, signals are discrete and finite so that the Discrete Fourier Transform is used. And its usage has been much popularized by the existence of the Fast Fourier Transform algorithm, making DFT computation quite accessible.