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

{L^p(\mathbb R)} (henceforth denoted by {L^p}) is the set of all square-integrable functions, i.e.,

\displaystyle L^p = \{ f : \mathbb R \rightarrow \mathbb R \vert \int_{\mathbb R} |f(x)|^p \;\text{d} x<\infty \}.

Note that {L^p} 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 {L^1} (i.e., integrable) function, we define the Fourier transform operator by

\displaystyle \mathcal F : L^1 \rightarrow \mathcal C_0(\mathbb R),

\displaystyle f(x) \longmapsto \widehat f(\xi) := \int_{\mathbb R} f(x) \exp{(2 \pi i x \xi)} \;\text{d} x .
It is clear the integral is well-defined a.e. {x\in\mathbb R}, in fact,

\displaystyle \|\mathcal F(f)\|_\infty \leq \|f\|_1.
However, that {\mathcal F(f)} eventually decays to zero as {|x|\rightarrow\infty} is not so obvious and the result and goes by the name of “Riemann-Lebesgue Lemma”.

Our objective is to define {\mathcal F} on {L^2} functions, because {L^2} is special and we can hope that {\mathcal F} is an operator on {L^2}. How nice it would be, if {\mathcal F} turned out to be a bijection, or better an isometry on {L^2}!

Just like the Fourier transform {f \mapsto \widehat f}, we can define the Inverse Fourier transform, {f \mapsto \mathcal G(f) = \check f} in the obvious way:

\displaystyle \mathcal G (f) = \check f(x) = \int_{\mathbb R} f(\xi) \exp{2 \pi i \xi x} \;\text{d} x.

Unfortunately, its not true that {\mathcal G(f) \in L^1} whenever {f} is. However, if {f} and {\widehat f} are both {L^1} functions, then

\displaystyle \mathcal G\circ \mathcal F (f) = -f \qquad \text{a.e. } \mathbb R.
Moreover, {f} is continuous except possibly at a measure-zero set.

The proof that {\mathcal G \circ \mathcal F (f) = -f} does not follow directly from Fubini, since

\displaystyle \mathcal G \circ \mathcal F (f) = \int_{\mathbb R}\int_{\mathbb R} f(y) \exp{-2 \pi i \xi y} \exp{2 \pi i \xi x} \;\text{d}y \;\text{d}\xi
need not be integrable. The trick is to bring in some good {L^1} function {g} whose FT and Inverse-FT exist and are equal and use the “change-of-hat” trick:

\displaystyle \int \hat f g = \int f \hat g, \qquad \forall f, g \in L^1.
If {g = \widehat g} and {f} are all in {L^1}, then it follows that {\widehat f} is in {L^1}. The function {g} has done its work and we can throw it away! But notice that the integrable function {g} whose FT is itself is, upto a scalar, the Gaussian {e^{-\pi x^2}}!

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

Theorem 1 If {f, \widehat f\in L^1} then {\check{\hat{f}} = -f } a.e., i.e. {\mathcal G\circ \mathcal F(f) = -f} a.e. and {f} is {\mathcal C^0} a.e. {\mathbb R}.

We now use the fact that {\mathcal C^\infty_0} functions (smooth functions decreasing to zero) are dense in {L^p} for {p <\infty} to define the FT on {L^2}.

Theorem 2 If {f} is smooth enough, then its FT {\widehat f} is also smooth of that order.

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

\displaystyle f \in \mathcal C^k \cap L^1\;\; \text{and} \;\; f^{(\alpha)} \in \mathcal C_0 \Rightarrow \widehat{f^{(\alpha)}}(\xi) = (2 \pi i \xi)^\alpha \widehat f(\xi)\quad \text{for } \alpha \leq k-1.

Now given an {f} in {L^2}, approximate it by functions {f_n} in {L^1 \cap L^2 \cap \mathcal C^\infty_0}. Since {\widehat{f_n} \in L^1 \cap \mathcal C^\infty_0}, by completeness of {L^2}, they will converge to some function. Call it {\widehat f}. Well-definedness of {\widehat f} is left to standard texts. But now, here comes the result we had secretly hoped for:

Theorem 3 (Plancherel) If {f \in L^1 \cap L^2} then {\widehat f\in L^2} and the Fourier transform {\mathcal F} is an isometry (bijective, continuous, norm-preserving, homeomorphism {\cdots}) onto its image and moreover,

{ \mathcal F|_{L^1 \cap L^2} } extends uniquely to {L^2}.

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 {p}-adic fields {\mathbb Q_p} 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.


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.