Generalized Sturmian Functions used for a discrete wavelet construction

Abstract

In this paper we discuss the implementation of scaling functions and wavelets based on Generalized Sturmian Functions (GSF). When dealing with finite dimensional spaces, the completeness relation of the GSF basis generates localized functions in two coordinates. By fixing one of the coordinates on particular points one defines the corresponding scaling functions. Wavelets are defined in a similar way by fixing, on a complementary set of functions, one of the coordinates on another set of particular points. This procedure allows for a multiscale decomposition of any signal. When the chosen points are the zeros of the th GSF, scaling functions generate a -dimensional space and happen to be orthogonal. We use GSF associated to the classical damped harmonic oscillator to build scaling functions and wavelets, and apply them to represent eye-tracking signals.