In this post, we shall outline a brief introduction to Fourier analysis.

(henceforth denoted by ) is the set of all square-integrable functions, i.e.,

Note that functions are defined upto a measure zero set, so are not functions but equivalence classes of functions that are equal except on a measure-zero set.

For every (i.e., integrable) function, we define the Fourier transform operator by

It is clear the integral is well-defined a.e. , in fact,

However, that eventually decays to zero as is not so obvious and the result and goes by the name of “Riemann-Lebesgue Lemma”.

Our objective is to define on functions, because is special and we can hope that is an operator on . How nice it would be, if turned out to be a bijection, or better an isometry on !

Just like the Fourier transform , we can define the Inverse Fourier transform, in the obvious way:

Unfortunately, its not true that whenever is. However, if and are both functions, then

Moreover, is continuous except possibly at a measure-zero set.

The proof that does not follow directly from Fubini, since

need not be integrable. The trick is to bring in some good function whose FT and Inverse-FT exist and are equal and use the “change-of-hat” trick:

If and are all in , then it follows that is in . The function has done its work and we can throw it away! But notice that the integrable function whose FT is itself is, upto a scalar, the Gaussian !

We now have a big tool with us, namely the Fourier Inversion theorem:

**Theorem 1** If then a.e., i.e. a.e. and is a.e. .

We now use the fact that functions (smooth functions decreasing to zero) are dense in for to define the FT on .

**Theorem 2** If is smooth enough, then its FT is also smooth of that order.

In proving this theorem, we use a fundamental property of FTs viz, it converts derivatives into polynomials:

Now given an in , approximate it by functions in . Since , by completeness of , they will converge to some function. Call it . Well-definedness of is left to standard texts. But now, here comes the result we had secretly hoped for:

**Theorem 3 (Plancherel)** If then and the Fourier transform is an isometry (bijective, continuous, norm-preserving, homeomorphism ) onto its image and moreover,

extends uniquely to .

**This is fantastic, because the uniqueness assures us that if two square-integrable functions share the same FT, then they must coincide almost everywhere!** Only the little theory developed so far can help us to solve some PDEs using Fourier Transforms. I discuss solving the transport equation using FTs in this post. Although I have to yet understand precisely how, I have learnt that the techniques of Fourier analysis help in solving problems in number theory. The -adic fields and the real field and their finite extensions are locally compact so we have a unique Haar measure that allows us to define Fourier transforms on them. I look forward to reading Ramakrishnan & Valenza’s *Fourier Analysis on Number Fields* soon.

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Taipiece: Prof. Athavale had taught this in a course at IIT Bombay. But I didn’t quite follow it then, since my understanding of real analysis was not so developed then. I once wrote an amusing post on him.

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