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The umbrella legitimization of many such Fourier transforms is as tempered distributions (where the sense of "distribution" is not the probability sense, but in the sense of Laurent Schwartz). The various "regularization" tricks amount to approaching the given distribution in the "weak *-topology" on distributions, by more tractable functions. Fourier transform on tempered distributions is (provably) continuous, so we conclude that all these trick must yield the same outcome.