Divergence of Spherical General Terms of Double Fourier Series

We prove the following theorem: For arbitrary there exists a nonnegative function such that and

The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Given an invertible matrix B and a finite or countable subset of , we consider the collection generating the closed subspace of . If that collection forms a frame for , one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.     

Spherical Perturbations of Schrödinger Equations

We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights and on , for which the a priori inequality holds with constant independent of . Here is the Laplace-Beltrami operator on the sphere . Due to the relation between and the tangential component of the gradient, , we obtain some "Morawetz-type" estimates for on . As a consequence we establish some new estimates for the free Schr?dinger propagator , which may be viewed as certain refinements of the -(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the dimensional Schr?dinger equation.     

Duality and Biorthogonality for Weyl-Heisenberg Frames

Let and let In this paper we investigate the relation between the frame operator and the matrix whose entries are given by for Here , for any We show that is bounded as a mapping of into if and only if is bounded as a mapping of into Also we show that if and only if where denotes the identity operator of and respectively, and Next, when generates a frame, we have that has an upper frame bound, and the minimal dual function can be computed as The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a generating a frame are inherited by In particular, we show that when generates a frame Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.     

Fourier Inequalities and Moment Subspaces in Weighted Lebesgue Spaces

For define where Pointwise estimates and weighted inequalities describing the local Lipschitz continuity of are established. Sufficient conditions are found for the boundedness of from into and a spherical restriction property is proved. A study of the moment subspaces of is next developed in the one-variable case, for locally integrable, a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in Characterizations are also given for each class. Applications related to the approximation and decomposition of are discussed.     

Numerical Integration over Spheres of Arbitrary Dimension

In this paper we study the worst-case error (of numerical integration) on the unit sphere for all functions in the unit ball of the Sobolev space where More precisely, we consider infinite sequences of m(n)-point numerical integration rules where: (i) is exact for all spherical polynomials of degree and (ii) has positive weights or, alternatively to (ii), the sequence satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) in has the upper bound where the constant c depends on s and d (and possibly the sequence This extends the recent results for the sphere by K. Hesse and I.H. Sloan to spheres of arbitrary dimension by using an alternative representation of the worst-case error. If the sequence of numerical integration rules satisfies an order-optimal rate of convergence is achieved.     

On the Problem of Optimal Reconstruction

We find lower bounds for linear and Alexandrov's cowidths of Sobolev's classes on Compact Riemannian homogeneous manifolds . Using these results we give an explicit solution of the problem of optimal reconstruction of functions from Sobolev's classes in .     

Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions

In this paper we develop a robust uncertainty principle for finite signals in which states that, for nearly all choices such that

Improved Approximation Algorithms for Geometric Set Cover

Given a collection S of subsets of some set and the set cover problem is to find the smallest subcollection that covers that is, where denotes We assume of course that S covers While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset and nondecreasing function f(·), there is a decomposition of the complement into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in and for guarding an x-monotone polygonal chain.     

Interassociates of the Free Commutative Semigroup on n Generators

The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·) denote this semigroup. We show that every interassociate can be written in the form , depending only on a n-tuple . Next, if and are isomorphic interassociates of (S, ·) such that , for x_ii and x_j in the generating set of S, then . Moreover, if and only if is a permutation of .     

Hardy Spaces for Laguerre Expansions

Let be the standard Laguerre functions of type a. We denote . Let and be the semigroups associated with the orthonormal systems and . We say that a function f belongs to the Hardy space associated with one of the semigroups if the corresponding maximal function belongs to . We prove special atomic decompositions of the elements of the Hardy spaces.     

Row Space Cardinalities

Let be the set of all Boolean matrices. Let R(A) denote the row space of , let , and let . By extensive computation we found that

Ideals of Ultrafilters on the Collection of Finite Subsets of an Infinite Set

Let J be an infinite set and let , i.e., I is the collection of all non empty finite subsets of J. Let denote the collection of all ultrafilters on the set I and let be the compact (Hausdorff) right topological semigroup that is the Stone-Cech Compactification of the semigroup equipped with the discrete topology. This paper continues the study of that was started in [3] and [5]. In [5], Koppelberg established that (where K( S) is the smallest ideal of a semigroup S) and for non empty she established . In this note, we show that for such that is infinite, is a proper subset of and , where .     

Old and New Morrey Spaces with Heat Kernel Bounds

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space of all locally integrable complex-valued functions f on such that for every open Euclidean ball B ⊂ with radius r_B there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying

Nonlinear Approximation by Trigonometric Sums

We investigate the -error of approximation to a function by a linear combination of exponentials on where the frequencies are allowed to depend on We bound this error in terms of the smoothness and other properties of and show that our bounds are best possible in the sense of approximation of certain classes of functions.     

On the Reproducing Formula of Calderon

Let and Under certain conditions on we shall prove that converges nontangentially to at for      

Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems

Let be a nontrivial probability measure on the unit circle the density of its absolutely continuous part, its Verblunsky coefficients, and its monic orthogonal polynomials. In this paper we compute the coefficients of in terms of the . If the function is in , we do the same for its Fourier coefficients. As an application we prove that if and if is a polynomial, then with and S the left-shift operator on sequences we have

Invariance Theorems in Approximation Theory and their Applications

Let B be a closed linear subspace of a Banach space F and let be a group of continuous linear operators , where G is a compact topological group. We prove that if is invariant under , then under some conditions on f, F, B, and G, there exists an element of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for polynomial approximation of and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of . Other examples and applications are discussed as well.     

Density, Overcompleteness, and Localization of Frames. I. Theory

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames and ( a discrete abelian group), relating the decay of the expansion of the elements of in terms of the elements of via a map . A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of , the relative measure of — both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements — and the density of the set in . Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.