Autocorrelation Functions as Translation Invariants in L1 and L2

The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions is a complete set of translation invariants for real-valued L¹ functions on a locally compact abelian group. It is shown here that there are proper subsets of that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset of is complete up to a sign. While stated here for functions on the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on and the integer lattice      

Orthogonal Algebraic Polynomial Schauder Bases of Optimal Degree

For any fixed we construct an orthonormal Schauder basis for C[-1,1] consisting of algebraic polynomials with The orthogonality is with respect to the Chebyshev weight.     

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.     

Generating Self-Map Monoids of Infinite Sets

Let be a countably infinite set, the group of permutations of , and the monoid of self-maps of . Given two subgroups , let us write if there exists a finite subset such that the groups generated by and are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to . Letting denote the obvious analog of for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups which are closed in the function topology on S, we have if and only if (as submonoids of E), and that for every subgroup (where denotes the closure of G in the function topology in S and its closure in the function topology in E).     

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.     

Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion where is the product Hardy space defined for example by R. Fefferman and is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in and respectively. In particular, we obtain a broad class of multipliers on and respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on and respectively.     

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

Uniformly Elliptic PDEs with Bounded,Measurable Coefficients

Let $L[\,\cdot\,]Let be a nondivergent linear second-order uniformly elliptic partial differential operator defined on functions with domain Consider the question, "When is a function u a solution of on ?" The naive answer, "u is a solution of on if and for all " is clearly too limited. Indeed, if the coefficients of L are in then L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. In this case there could be infinitely many functions that are "weak" but not classical solutions. More importantly, even if the coefficients of L are just bounded and measurable, the recent results of Krylov permit us to construct "solutions" of on and these "solutions" are generally no better than continuous; the "weak" solutions previously mentioned can be obtained by this construction, too. The preceding discussion provides us with an adequate extrinsic definition of solution (i.e., given a function u we either prove that it is or is not the result of such a construction) that has been used by several authors, but one that is not particularly satisfying or illuminating. Our major contribution in this paper is to show the following. I. There is an intrinsic definition of solution that is equivalent to the extrinsic one. II. Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.     

Weil Multipliers

This paper studies a class of linear operators on spaces of functions of one real variable, which correspond to multiplication by a measurable function under the Weil transform These operators are called Weil multipliers, and arise out of the authors' study of Gabor series and radar ambiguity functions. Representation theory provides a natural class of Weil multipliers: the set of doubly periodic functions with absolutely convergent Fourier series, It will be proved that functions in are multipliers for all and, therefore, define bounded linear endomorphisms of Also, we record the fact that the Wiener lemma tells us something about the orbit structure of these multipliers acting on function spaces on the Heisenberg nilmanifold. Linear maps that correspond to multiplication by a function under a unitary conjugacy have a particularly simple spectral decomposition, which yields an approximation theory for these operators and provides insight into the foundation of the authors' previous work on approximate orthonormal bases. Finally, the problem of inversion of a multiplier will be analyzed for smooth functions that have a specified structure near their zeros.     

Divergence of Spherical General Terms of Double Fourier Series

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

On the Reproducing Formula of Calderon

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

Nonperiodic Sampling of Bandlimited Functions on Unions of Rectangular Lattices

It is shown that a function is completely determined by the samples of on sets where and is irrational if and of If then the samples of on and only the first k derivatives of at 0 are required to determine f completely. Higher dimensional analogues of these results, which apply to functions and are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero and satisfying the rest. It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials that are not Riesz bases for A signal processing application in which such sampling sets arise naturally is described in detail.     

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.     

Multiresolution Analysis and Multivariate Approximation of Smooth Signals in CB(Rd

In this paper we derive rates of approximation for a class of linear operators on associated with a multiresolution analysis We show that for a uniformly bounded sequence of linear operators satisfying on the subspace a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider applications to extensions of generalized projection operators as well as to sampling series.     

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 .     

Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

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

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.     

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.