Fourier Transforms on the Line and in Space

Abstract

This chapter introduces a few features of the Fourier transform, mainly for its use in the design of Daubechies’ continuous, compactly supported, orthogonal wavelets. The principal use of the Fourier transform in the design of wavelets will consist in transforming “convolutions” of functions into ordinary multiplications of functions, which allow for an analysis easier than with convolutions. The material presented here involves mostly calculus, with only a few results with brief statements—such as Fubini’s theorem on the permutation of the order of multiple integrals [12, pp. 65-66], [21, pp. 384-388], [28, pp. 226-239]—borrowed from the literature in mathematical real analysis and complex analysis.