Time-Frequency Mean and Variance Sequences of Orthonormal Bases

We show that there exists an orthonormal basis for such that and are bounded sequences. We also show that there does not exist any orthonormal basis for , and being bounded sequences. This is motivated by a question posed by H.S. Shapiro on the mean and variance sequences associated to orthonormal bases.     

Affine Systems that Span Lebesgue Spaces

We establish rather weak conditions on under which the small scale affine system spans . The conditions are that the periodization of |ψ| be locally in L^p, that , and that the dilation matrices a_j are expanding, meaning . The periodization of ψ need not be constant; that is, the integer translates need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the , with the coefficients in the linear combination being local average values of f .     

Hermite Function Description of Feichtinger’s Space S0

For let be the set of all for which , where is the orthonormal base of Hermite functions. We show that if and only if and that if and only if . Here S₀ is Feichtinger’s space.     

From Multifractal Measures to Multifractal Wavelet Series

Given a positive locally finite Borel measure μ on R, a natural way to construct multifractal wavelet series is to set , where . Indeed, under suitable conditions, it is shown that the function F_μ inherits the multifractal properties of μ. The transposition of multifractal properties works with many classes of statistically selfsimilar multifractal measures, enlarging the class of processes which have self-similarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of F_μ are studied. As an application, multifractal Gaussian processes associated with F_μ are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arnéodo et al.     

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.     

Representations of Quantum Affinizations and Fusion Product

In this paper we study general quantum affinizations of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Presley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of q-characters give a ring structure * on the Grothendieck group of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).     

Direct Sum Decomposition of L2(Rn 1) into Subspaces Invariant under Fourier Transformation

Denote by the real-linear span of , where Under the concept of left-monogeneity defined through the generalized Cauchy-Riemann operator we obtain the direct sum decomposition of

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.     

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.     

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

Singularity of Vector Valued Measures in Terms of Fourier Transform

We study how the singularity (in the sense of Hausdorff dimension) of a vector valued measure can be affected by certain restrictions imposed on its Fourier transform. The restrictions, we are interested in, concern the direction of the (vector) values of the Fourier transform. The results obtained could be considered as a generalizations of F. and M. Riesz theorem, however a phenomenon, which have no analogy in the scalar case, arise in the vector valued case. As an example of application, we show that every measure from annihilating gradients of embedded in the natural way into i.e., such that for , has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results.     

Distributional Point Values and Convergence of Fourier Series and Integrals

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If and , and is locally integrable, then distributionally if and only if there exists k such that , for each a > 0, and similarly in the case when is a general distribution. Here means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by . We also show that under some extra conditions, as if the sequence belongs to the space for some and the tails satisfy the estimate ,\ as , the asymmetric partial sums\ converge to . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.     

A chain rule in $L^{1}\left({\operatorname*{div};\Omega}\right)$ and its applications to lower semicontinuity

A chain rule in the space is obtained under weak regularity conditions. This chain rule has important applications in the study of lower semicontinuity problems for general functionals of the form with respect to strong convergence in . Classical results of Serrin and of De Giorgi, Buttazzo and Dal Maso are extended and generalized.Received: 1 June 2002, Accepted: 7 January 2003, Published online: 6 June 2003     

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.     

Local André-Oort conjecture for the universal abelian variety

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of , the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety over may be defined over R. We call a point R-special if is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of , we show that an irreducible subvariety of containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.     

Affine pseudo-planes with torus actions

An affine pseudo-plane X is a smooth affine surface defined over which is endowed with an -fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic -action on X and X is an -surface, we shall show that the universal covering is isomorphic to an affine hypersurface in the affine 3-space and X is the quotient of by the cyclic group via the action where and It is also shown that a -homology plane X with and a nontrivial -action is an affine pseudo-plane. The automorphism group is determined in the last section.     

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.     

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.     

On the Reproducing Formula of Calderon

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

A Weighted Tauberian Theorem

We prove a Tauberian theorem of the form as where p(x) is a bounded periodic function and w(x) is a weighted function of power growth. It can be used to study the weighted average of the form