Non-MSF Wavelets From Six Interval MSF Wavelets

Abstract

In this article, we obtain two classes of non-MSF wavelets by considering two different classes of six-interval wavelet sets provided by Arcozzi, Behera and Madan (J. Geom. Anal.13 (2003) 557–579). Out of these classes one is countable, the non-MSF wavelets of which are non-MRA and the other one is uncountable, the non-MSF wavelets of which are MRA. The set of all non-MSF MRA wavelets of the latter class is shown to be pathconnected.

Minimally Supported Frequency (MSF) d-Dilation Wavelets

“…Employing Examples 2-4, we can construct non-MSF d-dilation wavelets with d > 1 for L 2 (R). The technique of constructing such wavelets is similar to the one utilized in [10,15,17]. Most of the constructed non-MSF d-dilation wavelets were non-MRA.…”

“…Wavelets and wavelet sets were extensively studied, considering various aspects such as the multiresolution analysis and the path connectivity of wavelets, by several researchers in the field of wavelets [10][11][12]14,15]. From a result by Chui and Shi [1], according to which for a dilation d such that d j / ∈ Q for all j ∈ N, the only wavelets that exist are MSF wavelets, the question of the existence or otherwise of a non-MSF wavelet for dilation other than the one considered by them arose.…”

“…Therefore, constructing non-MSF wavelets became a matter of interest. Following a procedure originated from the work of Bownik and Speegle [16], we construct non-MSF wavelets for L 2 (R) [10,15,17] and H 2 (R) [17] for dilation 2, and their association with multiresolution analysis was considered. The path connectivity of certain subsets of the set of wavelets was also discussed [15].…”

Non-MSF Mra Wavelets

“…Thus finding ways to create non-MSF wavelets has attracted various contributors in the field of wavelets. For instance, we refer to [2,4,10,11,12,13]. The purpose of this paper is to provide non-MSF wavelets in L 2 (R 2 ), considering specific wavelet sets in R 2 arose from an MRA wavelet sets in R and its scaling set.…”

Nonuniform N-multiwavelet sets and multiwavelet induced isomorphisms

“…This set W is known as a wavelet set in R. 4 Wavelet sets have been extensively studied in different perspectives by a large number of workers. [9][10][11][12][13][14][15][16][17][18][19][20] The notion of wavelet sets has been considered in higher dimensions and also for L 2 (G), where G is a locally compact Hausdorff topological group. 2,5,6,21 Considering Mallat's classical Multiresolution Analysis (MRA) which is based on the theory of spectral pairs, Gabardo and Nashed 8 generalized it to Nonuniform Multiresolution Analysis (NUMRA) associated with a spectrum comprising of a λ ∈ Λ r,N , k = 1, .…”