The genesis of convolution in Fourier analysis with applications

Abstract

This paper deals with derivation of the Fourier Integral Theorem, convolution theorem, dual convolution theorem, many classical and new algebraic and analytic properties of convolution, Poisson´s summation formula, integral representation of a band-limited signal, and Shannon´s sampling theorem. It is shown that the convolution originates in the context of partial sums of the Fourier series of a function f and the nth partial sum is expressed in terms of convolution of f and the Dirichlet kernel. The Cessaro and Abel means of a Fourier series of f are also expressed in terms of convolution of f and the Fejer and the Poisson kernels. It is shown that the Fourier inverse integral is Gauss, Abel and Cesaro summable to f(x) in the L¹-norm, and the pointwise summability theorems can be used to find the Fourier inversion formula almost everywhere in ℝ. Applications of convolution to the Fourier diffusion equation and the classical wave equation are presented.