Difference between ordinary and generalized functions
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I think many writers (including myself) use "generalized function" and "distribution" interchangeably, at least in rough terms, if only to emphasize the point that we don't so much care about the dual space to test functions because it's a dual space, but because the original space imbeds in it, is dense, and so the distributions are some sort of extension/generalization of "function".
Yes, there are hyperfunctions and other sorts of "generalized functions", although by an abuse of language we might call these "distributions".
A simple example is that Fourier transform cannot possibly map all (ordinary, but/and non-tempered) distributions to (ordinary) distributions, but we do know that Fourier transform maps test functions to the Paley-Wiener space, so Fourier transforms of (not-necessarily tempered) distributions can be defined, but only as in the dual of the Paley-Wiener space. For example, functionals like "evaluate at point zozo, which is off the real line" have no meaning for test functions nor Schwartz functions, but certainly do make sense on the Paley-Wiener space.
Yes, there are hyperfunctions and other sorts of "generalized functions", although by an abuse of language we might call these "distributions".
A simple example is that Fourier transform cannot possibly map all (ordinary, but/and non-tempered) distributions to (ordinary) distributions, but we do know that Fourier transform maps test functions to the Paley-Wiener space, so Fourier transforms of (not-necessarily tempered) distributions can be defined, but only as in the dual of the Paley-Wiener space. For example, functionals like "evaluate at point zozo, which is off the real line" have no meaning for test functions nor Schwartz functions, but certainly do make sense on the Paley-Wiener space.
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