The Path-Connectivity of s-Elementary Frame Wavelets with Frame MRA

Abstract

MRA wavelets have been widely studied in recent years due to their applications in signal processing. In order to understand the properties of the various MRA wavelets, it makes sense to study the topological structure of the set of all MRA wavelets. In fact, it has been shown that the set of all MRA wavelets (in any given dimension with a fixed expansive dilation matrix) is path-connected. The current paper concerns a class of functions more general than the MRA wavelets, namely normalized tight frame wavelets with a frame MRA structure. More specifically, it focuses on the parallel question on the topology of the set of all such functions (in the given dimension with a fixed dilation matrix): is this set path-connected? While we are unable to settle this general path-connectivity problem for the set of all frame MRA normalized tight frame wavelets, we show that this holds for a subset of it. An s-elementary frame MRA normalized tight frame wavelets (associated with a given expansive matrix A as its dilation matrix) is a normalized tight frame wavelet whose Fourier transform is of the form \(\frac{1}{\sqrt{2\pi}}\chi_{E}\) for some measurable set E⊂ℝd. In this paper, we show that for any given d×d expansive matrix A, the set of all (A-dilation) s-elementary normalized tight frame wavelets with a frame MRA structure is also path-connected.