Non-MSF Wavelets for the Hardy Space H²(R)

Abstract

Summary. All wavelets constructed so far for the Hardy space H 2 (R) are MSF wavelets. We construct a family of H 2 -wavelets which are not MSF. An equivalence relation on H 2 -wavelets is introduced and it is shown that the corresponding equivalence classes are non-empty. Finally, we construct a family of H 2 -wavelets with Fourier transform not vanishing in any neighbourhood of the origin.

An Equivalence Relation on Wavelets in Higher Dimensions

“…Later, in [3] and [11], examples of wavelets of L 2 (R) were constructed for each of these equivalence classes, by different methods. This equivalence relation was extended by the author to the wavelets of the Hardy space H 2 (R) in [2], where explicit examples of wavelets were constructed for each of the equivalence classes.…”

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.…”

Construction of Non-MSF Non-MRA Wavelets for L2(R) and H2(R) from MSF Wavelets

“…Recall that H 2 -wavelet sets with dilation 2 consisting of two intervals are given by (2) in the introduction. For r ∈ N and k = 2 l − 1, 1 ≤ l ≤ r, we have the following H 2 -wavelet sets with two intervals:…”

“…Further, they considered H 2 -wavelet sets [1,2,6,9] and characterized those H 2 -wavelet sets which have just one interval and also those with two intervals. Indeed, H 2 -wavelet sets consisting of two intervals are given by…”

“…Bownik and Speegle [3] characterized those dilations which admit non-MSF wavelets considering higher dimensional wavelets. Exploiting the structure of H 2 -wavelet sets having two intervals, Behera [2] constructed a family of non-MSF Hardy wavelets for H 2 (R) which, in addition, turns out to be a family of non-MRA Hardy wavelets for H 2 (R) due to Theorem 4.2, established in this paper.…”