I only just now discovered Wiener's work "Generalized Harmonic Analysis", which from what I understand was a generalization of Fourier Transform for functions that are not square-integrable.

However, we now know that we can take the generalized fourier transform of bounded signals to get distributions (generalized functions).

Wiener published his work in 1930, while according to Wikipedia Schwartz only introduced distribution theory in the late 1940s, so Wiener most likely didn't use them in his work. Is this correct?

And if so, is Wiener's work now obsolete because of generalized functions?

  • 1
    $\begingroup$ The idea looks quite the same. If $f$ doesn't decay fast enough as $x \to \infty$ then look at the Fourier transform of $g_n = f e^{- \pi x^2/n^2}$ whose FT is $\hat{f} \ast ne^{-\pi y^2n^2}$ which $\to \hat{f}$ in the sense of distributions as $n \to\infty$. Replacing the Gaussian by something else, as Wiener did, can be useful to relate the norm(s) of $\hat{g}_n$ with that of $f$ (which norms ? The Schwartz space considers infinitely many of them). So $f$ isn't a function but an operator acting of some class of functions. $\endgroup$ – reuns Jan 16 at 5:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.