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The Fourier transform is defined on $L^2$ by a density argument. It doesn't seem like it's constructive. So how would one go about computing the Fourier transform of a function in $L^2$ but not $L^1$? The only way I know of is to notice if the function is the Fourier transform of an $L^1$ function, and then use inversion.

user801923
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Since $f\chi_{[-n,n]}$ converges to $f$ in $L^2$, the Fourier transform of $f\in L^2(\mathbb R)$ is the $L^2$ limit of $$\int_{-n}^n e^{-2\pi i\xi t}f(t)\,dt,\quad n\to\infty \tag{1}$$ In practical computations, the limit is likely to exist pointwise; since we already know that it converges to $\hat f$ in $L^2$, the pointwise limit is $\hat f$.

Equation (1) is not really different from how we would find an improper integral of any kind. The main obstacle in the way of explicit computation is that (1) is hard to integrate.

Post No Bulls
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  • To increase the odds of (1) converging pointwise, one can use dyadic limits $-2^n\dots 2^n$, or something else adapted to the function at hand. – Post No Bulls Dec 06 '13 at 13:38