We show that the spin wavelets on the sphere $S^2$, which were constructed by the first author and Marinucci in an earlier article, can be chosen so as to form a nearly tight frame. These spin wavelets can be applied to the study of the polarization of cosmic microwave background radiation. For certain of these frames, there is a positive $C$ such that each frame element at scale $a^j$ is supported in a geodesic ball of radius $Ca^j$.
We propose a simple method to construct step mask and corresponding step wavelet functions that generate tight wavelet frames on the field of p-adic numbers. To construct tight wavelet frames we do not use the principle of unitary extension, we use Pontryagin's principle of duality.
Tight wavelet frames are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with tight wavelet frame construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to tight wavelet frames with high numbers of vanishing moments for all of the wavelet masks…
Wavelet frames for $L^2({\mathbb R})$ can be characterized by means of spectral techniques. This work uses spectral formulas to determine all the tight wavelet frames for $L^2({\mathbb R})$ with a fixed finite number of generators of minimal support. The method associates wavelet frames of this type with certain inner operator-valued functions in Hardy spaces. The cases with one and two generators are completely solved.
The question of existence is treated for near-minimizers for the distance functional (or $E$-functional in the interpolation terminology) that are stable under the action of certain operators. In particular, stable near-minimizers for the couple $(L^1, L^p)$ are shown to exist when the operator is the projection on wavelets and these wavelets possess only some weak conditions of decay at infinity.
Let $q\geq 2$ be an integer, and $\Bbb F_q^d$, $d\geq 1$, be the vector space over the cyclic space $\Bbb F_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E \subset \Bbb F_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2(\Bbb F_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for o…
We study the construction of nonuniform tight wavelet frames for the Lebesgue space $L^2(\mathbb{R})$, where the related translation set is not necessary a group. The main purpose of this paper is to prove the unitary extension principle (UEP) and the oblique extension principle (OEP) for construction of multi-generated nonuniform tight wavelet frames for $L^2(\mathbb{R})$. Some examples are also given to illustrate the results.
In this paper, an algorithm based on polyphase matrix for constructing a pair of orthogonal wavelet frames is suggested, and a general form for all orthogonal tight wavelet frames on local fields of positive characteristic is described. Moreover, we investigate their properties by means of the Fourier transform.
This work characterizes (dyadic) wavelet frames for $L^2({\mathbb R})$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated to the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis and extension principles, which are described here on the basis of the periodized Fourier transform. In a second paper of this series, we shall show how the spectral formulas obtained here permit us to calculate all the tight wavelet frames for $L^2({\mathbb…
We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization …
