### A Best Covering Problem.

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This paper obtains a class of tight framelet packets on ${L}^{2}\left({\mathbb{R}}^{d}\right)$ from the extension principles and constructs the relationships between the basic framelet packets and the associated filters.

The construction of nonseparable and compactly supported orthonormal wavelet bases of L 2(R n); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet...

We prove unconditionality of general Franklin systems in ${L}^{p}\left(X\right)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

We study the regularity of refinable functions by analyzing the spectral properties of special operators associated to the refinement equation; in particular, we use the Fredholm determinant theory to derive numerical estimates for the spectral radius of these operators in certain spaces. This new technique is particularly useful for estimating the regularity in the cases where the refinement equation has an infinite number of nonzero coefficients and in the multidimensional cases.

For a fusion Banach frame $(\{{G}_{n},{v}_{n}\},S)$ for a Banach space $E$, if $(\{{v}_{n}^{*}\left({E}^{*}\right),{v}_{n}^{*}\},T)$ is a fusion Banach frame for ${E}^{*}$, then $(\{{G}_{n},{v}_{n}\},S;\{{v}_{n}^{*}\left({E}^{*}\right),{v}_{n}^{*}\},T)$ is called a fusion bi-Banach frame for $E$. It is proved that if $E$ has an atomic decomposition, then $E$ also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.

A condition on a scaling function which generates a multiresolution analysis of ${L}^{p}\left({\mathbb{R}}^{d}\right)$ is given.

We prove a restriction theorem for the class-1 representations of the Heisenberg motion group. This is done using an improvement of the restriction theorem for the special Hermite projection operators proved in [13]. We also prove a restriction theorem for the Heisenberg group.

The aim of this paper is to obtain sharp estimates from below of the measure of the set of divergence of the m-fold Fourier series with respect to uniformly bounded orthonormal systems for the so-called G-convergence and λ-restricted convergence. We continue the study begun in a previous work.

This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author's contributions to the subject.

By the method of Rio [10], for a locally square integrable periodic function f, we prove $(f\left(\mu {\u2081}^{t}x\right)+...+f\left(\mu {\u2099}^{t}x\right))/n\to {\int}_{0}^{1}f$ for almost every x and t > 0.

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.

We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V₀ is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as in computing the period of a Riesz wavelet, answering in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the...

We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell}_{n}^{a}\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{L}_{n}^{a}\left(x\right)$, n = 0,1,2,..., in ${L}^{2}({\mathbb{R}}_{+},{x}^{a}dx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in {L}^{p}\left({x}^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.