Rational Franklin MRA and its Wavelets

Abstract

In signal processing, rational [Formula: see text]-wavelets are preferable than the wavelets corresponding to dyadic MRA because it allows more variations in scale factors of signal components. In this paper, for a rational number [Formula: see text] and [Formula: see text], we consider a collection [Formula: see text], the space of all continuous functions in [Formula: see text] that are linear on [Formula: see text] and [Formula: see text] for all [Formula: see text]. For [Formula: see text], under certain conditions, we prove that, if [Formula: see text] generates a [Formula: see text]-MRA, then [Formula: see text]. Also, we show that if [Formula: see text], there exists a function [Formula: see text], satisfying the above conditions, that generates [Formula: see text]-MRA. In addition, we construct orthonormal [Formula: see text]-wavelets corresponding to [Formula: see text]-MRA.