| Paley-wiener Theorem |
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| CATEGORIES ABOUT PALEY–WIENER THEOREM | |
| mathematical theorems | |
| generalized functions | |
| complex analysis | |
| fourier analysis | |
| hardy spaces | |
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Generally, the Fourier transform can be defined for any Tempered Distribution ; moreover, any distribution of compact support ''v'' is a tempered distribution. If ''v'' is a distribution of compact support and ''f'' is an infinitely differentiable function, the expression : is well defined. In the above expression the variable ''x'' in ''vx'' is a dummy variable and indicates that the distribution is to be applied with the argument function considered as a function of ''x''. It can be shown that the Fourier transform of ''v'' is a function (as opposed to a general tempered distribution) given at the value ''s'' by : and that this function can be extended to values of ''s'' in the complex space Cn. This extension of the Fourier transform to the complex domain is called the Fourier-Laplace Transform . Theorem. An entire function ''F'' on '''C'''n is the Fourier-Laplace transform of distribution ''v'' of compact support if and only if for all ''z'' ∈ '''C'''''n'', |
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