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Fourier transform of non-decaying functions

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Feb7-11, 01:10 AM
P: 625
if we consider a constant function [itex]f(x)=1[/itex], it is well-known that its Fourier transform is the delta function, in other words:

[tex]\int_{-\infty}^{+\infty}e^{-i\omega x}dx = \delta(\omega)[/tex]

The constant function does not tend to zero at infinity, so I was wondering: are there other functions that do not tend to zero at infinity but do have a Fourier transform?

I can think only of linear combinations of [tex]e^{-i\omega x}[/tex]. Are there others?
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Feb7-11, 04:27 PM
Sci Advisor
P: 6,035
I suspect that any bounded function would have an improper (including delta functions) Fourier transform.

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