On Franklin wavelets

Abstract

The Franklin wavelet (piecewise linear wavelet) is a wavelet function [Formula: see text] with dilation factor 2 which is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for all [Formula: see text]; which is constructed from a multiresolution analysis(MRA) with a scaling function [Formula: see text] which is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. Wavelets with composite dilation factor have the ability to produce long and narrow window functions, with varied orientations, well-suited to applications of image processing; also it is more effective and flexible for the construction of multiscale image representation. This motivates us to consider the following problem to construct some composite dilation Franklin wavelets. Given a natural number [Formula: see text], we ask, is there a continuous [Formula: see text] that is linear on [Formula: see text] and [Formula: see text], for [Formula: see text], [Formula: see text] and for all [Formula: see text], which generates an MRA with dilation factor [Formula: see text] to construct an orthonormal [Formula: see text]- wavelet, i.e. a wavelet with dilation factor [Formula: see text]? For [Formula: see text] and [Formula: see text] we have proved that [Formula: see text] generates an MRA with dilation factor [Formula: see text] only if [Formula: see text]. Next for [Formula: see text] and [Formula: see text], we have proved that, if [Formula: see text] generates an MRA with dilation factor [Formula: see text], then [Formula: see text]. Also, for [Formula: see text], we have proved existence of a continuous [Formula: see text] which generates an MRA with dilation factor [Formula: see text]. We have also constructed the corresponding orthonormal [Formula: see text]-wavelet.