Large classes of minimally supported frequency wavelets ofL 2(ℝ) andH 2(ℝ)

Abstract

We introduce a method to construct large classes of MSF wavelets of the Hardy space H 2 (R) and symmetric MSF wavelets of L 2 (R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L 2 (R) and H 2 (R). We also enumerate all symmetric wavelets of L 2 (R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H 2 (R). Finally, we construct families of MSF wavelets of L 2 (R) whose Fourier transform does not vanish in any neighbourhood of the origin.Notation. In this article, measure will always mean Lebesgue measure. All subsets of the real line we consider, are assumed to be Lebesgue measurable. With |E| we denote the Lebesgue measure of a measurable set E in R. We say that a relation between measurable sets holds almost everywhere (a.e.) if their characteristic functions are equal a.e. Thus, A = B a.e. means that χ A = χ B a.e., and n A n = A a.e. means that n χ An = χ A a.e., where denotes the disjoint union.

Dimension functions, scaling sequences, and wavelet sets

“…There are many papers in the literature devoted to various aspects of wavelet sets: their properties, interrelations with other relevant objects, construction methods, etc. We refer the reader to [2], [6], [13], [14], [18] and references therein.…”

“…Finally, by Theorem 1.9 of [7], if (v j ) is a sequence of functions in L 2 (T) satisfying v j (ξ) = 0 for all ξ ∈ Ω j and all j, and the above conditions (1) and (2), then the function ψ defined byψ(2ξ)…”

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

“…An orthonormal wavelet whose Fourier transform has the support of smallest possible measure is called a minimally supported frequency (MSF) wavelet. In fact, an MSF wavelet ψ is a wavelet which is associated with a wavelet set W in the sense that the support of ψ is W [1,[4][5][6][7][8][9][10]. One of the earliest wavelets, namely Shannon wavelet for dilation 2, has W = [−2π, −π] ∪ [π, 2π] as its wavelet set, which is a union of two disjoint intervals of R. Wavelet sets in R which are unions of two disjoint intervals and also those which are unions of three disjoint intervals have been characterized by Ha, Kang, Lee and Seo [6].…”

“…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…”

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

“…They also characterized all H 2 -wavelet sets consisting of two disjoint intervals. In [3] (see also [1]) we proved a result on the structure of H 2 -wavelet sets consisting of a finite number of intervals and, as an application, characterized 3-interval H 2 -wavelet sets. All these wavelet sets depend on a finite number of integral parameters, which proves that there are countably many H 2 -wavelet sets which are unions of at most three disjoint intervals.…”