Tempered Distributions and Fourier Transforms
Tempered distributions, as functionals, are continuous with respect to a slightly weaker mode of convergence than that described in Section 2.4, hence are slightly milder, as a class, than the class of all Schwartzian distributions. The mildness is not a local matter (the 03B4 function and all its derivatives are tempered distributions), but has to do with the behavior as ∥x∥ → ∞. The Fourier transform of a tempered distribution is easily defined; it is also a tempered distribution.
KeywordsThe class S of test functions that decrease rapidly at tempered distributions growth at ∞; a tempered distribution as some derivative of a continuous function of slow growth Fourier transforms in S inversion by Fejér’s method Fourier transforms of tempered distributions power spectrum of a perpetually oscillating function
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