In this paper, we provide a family of compactly supported orthonormal complex wavelets with dilation 4 such that the generating wavelet functions are symmetric/antisymmetric and have linearly increasing orders of vanishing moments and smoothness.

We construct smooth nonseparable compactly supported refinable functions that generate multiresolution analyses on L2(R), d > 1. Using these refinable functions we construct smooth nonseparable compactly supported orthonormal wavelet systems. These systems are nonseparable, in the sense that none of its constituent functions can be expressed as the product of two functions defined on lower dime...

Two simple constructive methods are presented to compute compactly supported tight wavelet frames for any given refinable function whose mask satisfies the QMF or sub-QMF conditions in the multivariate setting. We use one of our constructive methods in order to find tight wavelet frames associated with multivariate box splines, e.g., bivariate box splines on a three or four directional mesh. Mo...

Let Ω ⊆ R be a non-empty open set for which the Sobolev embedding H 0 (Ω) −→ L(Ω) is compact, and let V ∈ L∞(Ω) be a potential taking only positive real values and satisfying the asymptotics V (·) 〈·〉 for some α ∈ ]3,∞[. We establish the discreteness of the set of real transmission eigenvalues for both Schrödinger and Helmholtz scattering with these potentials.

This paper considers Gibbs’ phenomenon for scaling vectors in L2(R). We first show that a wide class of multiresolution analyses suffer from Gibbs’ phenomenon. To deal with this problem, in [11], Walter and Shen use an Abel summation technique to construct a positive scaling function Pr, 0 < r < 1, from an orthonormal scaling function φ that generates V0. A reproducing kernel can in turn be con...

T2* and quantitative susceptibility mapping (QSM) of magnetic resonance imaging (MRI) images provide different type inner structure information of scanned organs. If they can be properly fused into one set, the details of the scaned organ can be revealed more clearly. In this paper, a 3D MRI image fusion method based on 3D compactly supported shearlet transform (3D-CSST) and 3D dual tree compac...

For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (or...

This article extends results on regular implementablity in [3] and [8] to the case when the signal space is not an injective cogenerator, for instance, the space D of compactly supported smooth functions on R. In this case the bijective correspondence between behaviors and modules fails to hold; also projections and sums of behaviors need not in general be behaviors. A more general version of i...

In this article we obtain estimates on the approximate eigenstructure of channels with a spreading function supported only on a set of finite measure |U |. Because in typical application like wireless communication the spreading function is a random process corresponding to a random Hilbert–Schmidt channel operator H we measure this approximation in terms of the ratio of the p–norm of the devia...

To effectively capture singularities in high-dimensional data and functions, multivariate compactly supported tight framelets, having directionality and derived from refinable box splines, are of particular interest in both theory and applications. The d-dimensional Haar refinable function χ[0,1]d is a simple example of refinable box splines. For every dimension d ∈ N, in this paper we construc...