Riesz bases in L2(0,1) related to sampling in shift-invariant spaces

AbstractThe Fourier duality is an elegant technique to obtain sampling formulas in Paley–Wiener spaces. In this paper it is proved that there exists an analogue of the Fourier duality technique in the setting of shift-invariant spaces. In fact, any shift-invariant space Vφ with a stable generator φ is the range space of a bounded one-to-one linear operator T between L2(0,1) and L2(R). Thus, regular and irregular sampling formulas in Vφ are obtained by transforming, via T, expansions in L2(0,1) with respect to some appropriate Riesz bases

Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis

Linear combinations of generators in multiplicatively invariant spaces

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega,\mathcal{H})$ that are invariant under multiplications of (some) functions in $L^{\infty}(\Omega)$. In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in $L^2(\mathbb{R}^d)$.Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematic