A Non-linear Approach to Signal Processing by Means of Vector Measure Orthogonal Functions

Abstract

Sequences of real functions that are orthogonal with respect to a vector measure are a natural generalization of orthogonal systems with respect to a parametric measure. In this paper we develop a new procedure to construct non-linear approximations of functions by defining orthogonal series in spaces of square integrable functions with respect to a vector measure whose Fourier coefficients are also functions. We study the convergence properties of such series, defining a convenient approximation procedure for signal processing involving time dependence of the measure. Some examples involving classical orthogonal polynomials are given.