Orthogonal Polynomials of Discrete Variable and Boundedness of Dirichlet Kernel

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal expansions with respect to L^∞ norm, which generalize analogous results obtained, for little q-Legendre, little q-Jacobi, and little q-Laguerre polynomials, by the authors.     

Continuous and Discrete Tight Frames of Orthogonal Polynomials for a Radially Symmetric Weight

This paper considers tight frame decompositions of the Hilbert space ℘ _n of orthogonal polynomials of degree n for a radially symmetric weight on ℝ ^d , e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘ _n with the property that each f∈℘ _n can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p _ξ obtained from p by moving its pole to ξ∈S:={ξ∈ℝ ^d :|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for ℘ _n .      

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system and a sequence of nonzero numbers,let be the linear operator defined on the linear spaceof all polynomials via for all .We investigate conditions on and under which can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for , a generalized Laguerre polynomial system, no can simultaneously preserve the orthogonality of twoadditional Laguerre systems, and , where and . On the other hand, for ,the Chebyshev polynomial system and , simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.     

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

Mathematical Notes - Let there be given a partition of the closed interval $$[-1,1]$$ by arbitrary nodes $${eta_j}_{j=0}^N$$ , where $$lambda_N=max_{0le j le N-1} (eta_{j+1}-eta_{j})$$ ....     

基于再生核三次样条基底的构造及其应用

将三次样条理论与再生核理论相结合,利用再生核函数巧妙地构造了三次样条函数空间的一组基底.基于三次样条插值的高收敛特点,得到了微分方程边值问题近似解的一种新的求解方法.数值算例展现出算法简单、有效.     

The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials I: Mellin Transform Techniques

The Fourier transform of orthogonal polynomials with respect to their own orthogonality measure defines the family of Fourier–Bessel functions. We study the asymptotic behaviour of these functions and of their products, for large real values of the argument. By employing a Mellin analysis we construct a general framework to exhibit the relation of the asymptotic decay laws to certain dimensions of the orthogonality measure, that are defined via the divergence abscissas of suitable integrals. The unifying r?le of Mellin transform techniques in deriving classical and new results is underlined. Submitted: November 5, 2004. Accepted: January 6, 2006.     

On the Convergence of Fourier Series with Orthogonal Polynomials inside and on the Closure of a Region

We study the rate of convergence of Fourier series of orthogonal polynomials over an area inside and on the closure of regions of the complex plane.     

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation,Factorization and Solutions

We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases

We obtain the spectral decomposition of the hypergeometric differential operator on the contour Re z = 1/2. (The multiplicity of the spectrum of this operator is 2.) As a result, we obtain a new integral transform different from the Jacobi (or Olevskii) transform. We also construct an ₃ F ₂-orthogonal basis in a space of functions ranging in ℂ². The basis lies in the analytic continuation of continuous dual Hahn polynomials with respect to the index n of a polynomial.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 31–46, 2005Original Russian Text Copyright © by Yu. A. Neretin     

Spectral Measures and Jacobi Matrices Related to Laguerre-Type Systems of Orthogonal Polynomials

This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality. Date received: September 7, 1995. Date revised: April 17, 1996.     

The Approximation of Functions by Partial Sums of the Fourier Series in Polynomials Orthogonal on Arbitrary Grids

Russian Mathematics - for arbitrary continuous function f(t) on the segment [?1,1] we construct discrete Fourier sums Sn,N(f,t) on a system of polynomials forming an orthonormal system on...     

一类积分边值问题的再生核数值算法

研究了一类具有积分条件的边值问题.基于再生核理论,巧妙地构造了一个具有积分条件的再生核空间,并且给出了再生核函数表达式.应用泛函分析中的算子理论及逼近论思想,给出了方程的近似解,即再生核数值算法.通过实例验证了算法的可行性和有效性.     

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.     

The Difference Between a Discrete and Continuous Harmonic Measure

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let \(\omega _h(0,\cdot ;D)\) be the discrete harmonic measure at \(0\in D\) associated with this random walk, and \(\omega (0,\cdot ;D)\) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function \(\sigma _D(z)\) on \(\partial D\) such that for functions g which are in \(C^{2+\alpha }(\partial D)\) for some \(\alpha >0\) we have

$$\begin{aligned} \lim _{h\downarrow 0} \frac{\int _{\partial D} g(\xi ) \omega _h(0,|\mathrm{d}\xi |;D) -\int _{\partial D} g(\xi )\omega (0,|\mathrm{d}\xi |;D)}{h} = \int _{\partial D}g(z) \sigma _D(z) |\mathrm{d}z|. \end{aligned}$$

We give an explicit formula for \(\sigma _D\) in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

     

Unimodular Triangulations and Coverings of Configurations Arising from Root Systems

Existence of a regular unimodular triangulation of the configuration , where ⁺ is the collection of the positive roots of a root system and where (0, 0,...,0 ) is the origin of , will be shown for = B _n , C _n , D _n and BC _n . Moreover, existence of a unimodular covering of a certain subconfiguration of the configuration A _n+1 ⁺ will be studied.     

波动方程的解与再生核空间的关系

从线性变换入手,应用再生核空间理论,建立了波动方程的解与再生核空间的关系.得到了波动方程的解的反演公式及等距等式,为再生核理论的应用提供了新的思路.     

The Convergence Rate of Block Preconditioned Systems Arising from LMF-based ODE Codes

The solution of ordinary an partial differential equations using implicit linear multi-step formulas (LMF)is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systems x = b. Block-circulant preconditioners have been propose to solve these linear systems. By investigating the spectral condition number of , we show that the conjugate gradient method, when applied to solving the normalize preconditioned system, converges in at most O(log s) steps, where the integration step size is O(1/s). Numerical results are given to illustrate the effectiveness of the analysis.This revised version was published online in October 2005 with corrections to the Cover Date.     

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials

This work concerns superharmonic perturbations of a Gaussian measure given by a special class of positive weights in the complex plane of the form w(z) = exp(−|z|² + U^μ(z)), where U^μ(z) is the logarithmic potential of a compactly supported positive measure μ. The equilibrium measure of the corresponding weighted energy problem is shown to be supported on subharmonic generalized quadrature domains for a large class of perturbing potentials U^μ(z). It is also shown that the 2 × 2 matrix d-bar problem for orthogonal polynomials with respect to such weights is well-defined and has a unique solution given explicitly by Cauchy transforms. Numerical evidence is presented supporting a conjectured relation between the asymptotic distribution of the zeroes of the orthogonal polynomials in a semi-classical scaling limit and the Schwarz function of the curve bounding the support of the equilibrium measure, extending the previously studied case of harmonic polynomial perturbations with weights w(z) supported on a compact domain. Submitted: July 25, 2008. Accepted: October 1, 2008. Work supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche sur la nature et les technologies du Québec (FQRNT).