Tight frames of exponentially decaying Wannier functions

Tight frames of exponentially decaying Wannier functions

Let L be a Schrödinger operator ( i ∂ ∂x −A(x))2+V (x) with periodic magnetic and electric potentials A,V , a Maxwell operator ∇× 1 ε(x)∇× in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R with coefficients periodic with respect to a lattice Γ. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We cons...

Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

Regularity of Refinable Functions with Exponentially Decaying Masks

The smoothness property of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the smoothness properties of refinable functions with exponentially decaying masks and an isotropic ...

Constructing tight frames of multivariate functions

The paper presents a method of construction of tight frames for L(Ω), Ω ⊂ R. The construction is based on local orthogonal matrix extension of vectors associated with the transition matrices across consecutive resolution levels. Two explicit constructions are given, one for linear splines on triangular polygonal surfaces with arbitrary topology and the other for quadratic splines associated wit...

Jost Functions for Jacobi Operators with Super-exponentially Decaying Parameters

The decay of the parameters {an, bn} for a Jacobi operator J on `(N) is related to the analyticity of the Jost function u(z; J) associated with J , which is in turn related to the spectral measure dμ of J . Damanik and Simon demonstrated the equivalence between the exponential decay of these parameters and the analyticity of the Jost function on a disk whose radius is given by the rate of decay...