Parseval frame wavelets associated with A-FMRA

Abstract

Mohamed El Naschie’s E-infinity theory has introduced a new framework for understanding and describing nature that is resolution dependent. Wavelets and multiresolution analysis are good mathematical tools to support EI Naschie’s picture of the resolution dependence of the observations. In this paper, inspired by these theory and applications, we study Parseval frame wavelets (PFWs) in L 2( R n ) with matrix dilations of the form ( Df ) ( x ) = 2 f ( Ax ) , where A is an arbitrary n × n expanding matrix with integer coefficients, such that ∣det A∣ = 2. We prove that all PFWs associated to A-FMRA are equivalent to semi-orthogonal Parseval frame wavelets, and characterize all PFWs associated to A-FMRA by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued in L 2( R n ). Then, we discover the relation between the spectrum of the central space of an A-FMRA and the supported set of bracket function of its generator and obtain a characterization of PFWs associated with an A-FMRA by the spectrum of the central space of an FMRA. In each section, we construct concrete examples. Thus, we give some mathematical methods to support El Naschie’s picture of the resolution dependence of the observations.