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Advanced Mathematics for Applications by Andrea Prosperetti

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10

The Fourier and Hankel Transforms

In this chapter the Fourier and Hankel transforms are presented in a rather formal way without attempting to provide rigorous justifications of the various statements. The reason is that stringent hypotheses make proofs relatively straightforward, and these can be found in a large number of texts. The Fourier transform, however, is a much more powerful tool than these simplified results imply, and the general proofs are more difficult; a useful resource is the book by Champeney (1987). The Fourier transform of generalized functions is treated in §20.10 p. 595.

10.1 Heuristic motivation

According to §9.9 of the previous chapter, a function f having bounded variation in the interval −L < xL can be ...

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