Principles of Fourier Analysis, 2nd Edition

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Gaussians as Test Functions and Proofs of Important Theorems

There are two major reasons for this chapter. One is to finally confirm those few important theorems that we’ve been using with only a promise of proof “sometime in the future”: the fundamental theorem on invertibility, the main theorems on the differentiation identities, the second theorem on the fundamental identity and the second theorem on the convolution identities. The other is to introduce some basic concepts that will later lead to a much more general and powerful theory of Fourier analysis. In particular, you should note the “Gaussian test for equality” in the next section and the way we use it with the fundamental identity to prove the theorems on invertibility and differential ...