On MRA Riesz wavelets

On a Conjecture about Mra Riesz Wavelet Bases

Let φ be a compactly supported refinable function in L2(R) such that the shifts of φ are stable and φ̂(2ξ) = â(ξ)φ̂(ξ) for a 2π-periodic trigonometric polynomial â. A wavelet function ψ can be derived from φ by ψ̂(2ξ) := e−iξ â(ξ + π)φ̂(ξ). If φ is an orthogonal refinable function, then it is well known that ψ generates an orthonormal wavelet basis in L2(R). Recently, it has been shown in the liter...

Characterization of Riesz bases of wavelets generated from multiresolution analysis

We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a co...

p-adic refinable functions and MRA-based wavelets

A Class of Non-mra Band-limited Wavelets

We give a characterization of a class of band-limited wavelets of L(R) and show that none of these wavelets come from a multiresolution analysis (MRA). For each n ≥ 2, we construct a subset Sn of R which is symmetric with respect to the origin. We give necessary and sufficient conditions on a function ψ ∈ L(R) with supp ψ̂ ⊆ Sn to be an orthonormal wavelet. This result generalizes the characteri...

Small Support Spline Riesz Wavelets in Low Dimensions

In [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556], a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in [B. Han and Z. Shen, J. Fourier Anal. Appl., 11 (2005), 615–637]. Motivated by these two papers, we develop in this article a general theory and a construction meth...