Nonnegative Trigonometric Polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szegő is under consideration. An analog of the Fejér—Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szegő are obtained explicitly. Associated cosine polynomials k _n (θ) are constructed in such a way that { k _n (θ) } are summability kernels. Thus, the L _p , pointwise and almost everywhere convergence of the corresponding convolutions, is established. April 26, 2000. Date revised: December 28, 2000. Date accepted: February 8, 2001.     

On Approximation by Ridge Functions

Ridge functions are defined as functions of the form , where , belongs to the given ``direction' set . In this paper we study the fundamentality of ridge functions for variable directions sets A and discuss the rate of approximation by ridge functions. Date received: June 7, 1994. Date revised: August 3, 1995.     

Characterising Pairs of Refinable Splines

A complete characterisation is given, in terms of Fourier transforms, of pairs of refinable univariate spline functions, with knots at the integers, whose integer translates form a Riesz basis.     

Regularity of Multivariate Refinable Functions

The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A , although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on the spectral properties of A . September 30, 1996. Dates revised: December 1, 1996; February 14, 1997; August 1, 1997; November 11, 1997. Date accepted: November 14, 1997.     

On the Norm of the Fourier—Gegenbauer Projection in Weighted L p Spaces

We extend the results of Pollard [4] and give asymptotic estimates for the norm of the Fourier—Gegenbauer projection operator in the appropriate weighted L _p space. In particular, we settle the question of whether the projection is bounded for p=(2λ+1)/λ and p=(2λ+1)/(λ+1) , where λ is the index for the family of Gegenbauer polynomials under consideration. March 19, 1997. Date revised: June 3, 1998. Date accepted: August 1, 1998.     

Frames and oversampling formulas for band limited functions

In this article, we obtain families of frames for the space B _ω of functions with band in [?ω, ω] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of Ron and Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates of a finite number of functions (generators) to be a frame or a Riesz basis for B _ω . We also give explicit formulas for the dual generators, and we apply them to Hilbert transform sampling and derivative sampling. Finally we provide numerical experiments that support the theory.     

The Nikol'skii—Timan—Dzjadyk Theorem for Functions on Compact Sets of the Real Line

The Nikol'skii—Timan—Dzjadyk theorem concerning the polynomial approximation of functions on the interval [-1,1] is generalized to the case of the approximation of functions given on a compact set on the real line which can consist of an infinite number of intervals. December 8, 1999. Date revised: August 15, 2000. Date accepted: October 19, 2000.     

多频率多函数小波

本文把通过方向多分辨分析构造的由一个函数生成的多频率小波推广到由有限个函数生成的多频率小波，给出由n2j1 j2个函数φ1,…φn,ψ1,…ψn(2j1 j2-1)的平移生成Vj(l)空间的Riesz基的充分必要条件．     

多频率小波

通过方向多分辨分析把由一个函数生成的多频率小波推广到由有限个函数生成的多频率小波，给出由函数φ1,…，φn，…ψn(2^j1 ^j2-1)的平移生成Vj(1)空间的Riesz基的充分必要条件，同时给出该小波的分解式。     

On the Riesz constants for systems of integer translates

In this paper, one-parameter families of integer translates of the Gaussian and Lorentz functions are studied. For a Lorentz function, we obtain formulas for the coefficients of the series defining node functions and show that the limit value of node functions is given by a sample function. For systems of translates generated by the Gaussian and Lorentz functions as well as by the node functions related to them, we obtain explicit expressions for the Riesz constants and study the parameter-dependent behavior of these constants. While proving some of the results of this paper, we establish the monotonicity of a special ratio of two Jacobi theta functions, a fact which is of interest in itself.     

A note on fast Fourier transforms for nonequispaced grids

In this paper, we are concerned with fast Fourier transforms for nonequispaced grids. We propose a general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the polynomials by special linear combinations of translates of suitable functions ϕ. We derive estimates for the approximation error. In particular, we improve the estimates given by Dutt and Rokhlin [7]. As a practical consequence, we obtain a criterion for the choice of the parameters involved in the fast transforms. This revised version was published online in August 2006 with corrections to the Cover Date.     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

We consider approximation of L ^p functions by Hardy functions on subsets of the circle for . After some preliminaries on the possibility of such an approximation which are connected to recovery problems of the Carleman type, we prove existence and uniqueness of the solution to a generalized extremal problem involving norm constraints on the complementary subset. December 6, 1995. Date revised: August 26, 1996.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

Boundedness of Commutators Generated by Campanato-type Functions and Riesz Transforms Associated with Schrdinger Operators

Let■=-△+V be a Schrdinger operator on R~n,n3,where △is the Laplacian on R~n and V≠0 is a nonnegative function satisfying the reverse Holder's inequality.Let[b,T]be the commutator generated by the Campanatotype function b∈■ and the Riesz transform associated with Schrdinger operator T=▽(-△+V)~(-1/2).In the paper,we establish the boundedness of[b,T]on Lebesgue spaces and Campanato-type spaces.     

Riesz Systems and Controllability of Heat Equations with Memory

The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target.

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This paper fits into the research program of the GNAMPA-INDAM .     

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms ||⋅|| _Hs([0,1]) for s∈ (-0.824926,2.5) . In particular, the multiwavelets form Riesz bases for L ₂ ([0,1]) . February 2, 1998. Date revised: February 19, 1999. Date accepted: March 5, 1999.     

Nonnegative Linearization for Polynomials Orthogonal with Respect to Discrete Measures

We give conditions for the coefficients in three term recurrence relations implying nonnegative linearization for polynomials orthogonal with respect to measures supported on convergent sequences of points. The previous methods were unable to cover this case. January 14, 1999. Date revised: January 31, 2000. Date accepted: October 24, 2000.     

Dependence Relations Among the Shifts of a Multivariate Refinable Distribution

Refinable functions are an intrinsic part of subdivision schemes and wavelet constructions. The relevant properties of such functions must usually be determined from their refinement masks. In this paper, we provide a characterization of linear independence for the shifts of a multivariate refinable vector of distributions in terms of its (finitely supported) refinement mask. March 14, 1998. Dates revised: February 3, 1999 and August 6, 1999. Date accepted: November 16, 1999.