Convergence of Fourier Transforms and their Inverses

Fourier transforms of non-periodic functions are defined as integrals from $-\infty$ to $+\infty$ . What does it mean for an integral to be taken from $-\infty$ to $+\infty$ ?

By definition, $latex \int\limits_{-\infty}^{\infty} f\, dt=
\lim\limits_{T\to\infty}\int\limits_{-T}^{T} f\, dt$

(Strictly speaking, the right side should be $\lim\limits_{M\to -\infty,N\to\infty}\int\limits_{M}^{N} f\, dt$ . But, to illustrate our point, we assume this is same as the right side).

So, whenever, we talk about integrals from $-\infty$ to $\infty$ , we are implicitly talking about convergence of a limit.

Now, define $F_n(t) = \int\limits_{-n}^{n} f(\gamma)e^{-it\gamma} \,d\gamma$ . Now, when we say $\int\limits_{-\infty}^{\infty} f(\gamma)e^{-it\gamma}\,d\gamma$ exists and is equal to $F(t)$ , we are essentially saying $F_n(t)\to F(t)$ as $n\to \infty$ . That is not all. Remember, we are talking about limits of sequences of functions, not of sequences of numbers. So, immediately, we have the question, what type of convergence we are talking about? Is it point-wise convergence, uniform convergence, convergence in $L^p(\mathbb{R})$ or something else?

All these points surface when we deal with Fourier transforms and inverse Fourier transforms of functions. The Fourier transforms and inverse Fourier transforms behave differently for different kinds of functions (functions in $L^1(\mathbb{R}), L^2(\mathbb{R}$ , etc).

In this article, we will discuss these aspects.