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

### Luis M. García-Raffi

Universidad Politécnica de Valencia, Spain### E. Jiménez Fernández

Universidad Politécnica de Valencia, Spain### Enrique A. Sánchez Pérez

Universidad Politécnica de Valencia, Spain

## 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.

## Cite this article

Luis M. García-Raffi, E. Jiménez Fernández, Enrique A. Sánchez Pérez, A Non-linear Approach to Signal Processing by Means of Vector Measure Orthogonal Functions. Publ. Res. Inst. Math. Sci. 49 (2013), no. 2, pp. 241–269

DOI 10.4171/PRIMS/105