In this paper we construct Powell–Sabin spline multiwavelets on the hexagonal lattice in a shift-invariant setting. This allows us to use Fourier techniques to study the range of the smoothness parameter s for which the wavelet basis is a Riesz basis in the Sobolev space H(R), and we find that 0.431898 < s < 5/2. For those s, discretizations of H -elliptic problems with respect to the wavelet b...

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also symmetric, there exists a Riesz basis exponential functions in the space $L^2(\Omega)$. The result new $d$ greater than one.

In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective ...

Compactly supported Riesz wavelets are of interest in several applications such as image processing, computer graphics and numerical algorithms. In this paper, we shall investigate compactly supported MRA Riesz multiwavelet bases in L2(R). An algorithm is presented to derive Riesz multiwavelet bases from refinable function vectors. To illustrate our algorithm and results in this paper, we prese...

In this article we suppose that E is an ordered Banach space the positive cone of which is defined by a countable familyF={fi|i ∈ N} of positive continuous linear functionals of E, i.e. E+ = {x ∈ E | fi(x) ≥ 0, for each i} and we study the existence of positive (Schauder) bases in the ordered subspaces X of E with the Riesz decomposition property. So we consider the elements x of E as sequences...

Let G be a countably infinite group of unitary operators on a complex separable Hilbert space H. Let X = {x1, ..., xr} and Y = {y1, ..., ys} be finite subsets of H, r < s, V0 = spanG(X), V1 = spanG(Y ) and V0 ⊂ V1. We prove the following result: Let W0 be a closed linear subspace of V1 such that V0 ⊕ W0 = V1 (i.e., V0 + W0 = V1 and V0 ∩ W0 = {0}). Suppose that G(X) and G(Y ) are Riesz bases for...

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications t...

In this article we obtain families of frames for the space Bω of functions with band in [−ø, ø] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates o...

We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also symmetric, there exists a Riesz basis exponential functions in the space $L^2(\Omega)$. The result new $d$ greater than one.