Fourier Transforms and Their Applications

This chapter deals with Fourier transforms in L1(ℝ) and in L2 (ℝ) and their basic properties. Special attention is given to the convolution theorem and summability kernels including Cesáro, Fejér, and Gaussian kernels. Several important results including the approximate identity theorem, general Parseval’s relation, and Plancherel theorem are proved. This is followed by the Poisson summation formula, Gibbs’ phenomenon, the Shannon sampling theorem, and Heisenberg’s uncertainty principle. Many examples of applications of the Fourier transforms to mathematical statistics, signal processing, ordinary differential equations, partial differential equations, and integral equations are discussed. Included are some examples of applications of multiple Fourier transforms to important partial differential equations and Green’s functions.