Polynomial operators and local smoothness classes on the unit interval

We obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded.     

Uniform Convergence of Fourier–Jacobi Series

Necessary and sufficient conditions which imply the uniform convergence of the Fourier–Jacobi series of a continuous function are obtained under an assumption that the Fourier–Jacobi series is convergent at the end points of the segment of orthogonality [−1,1]. The conditions are in terms of the modulus of continuity, Λ-variation, and the modulus of variation of a function.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

基于Jacobi多项式零点的Grünwald插值算子

本文考虑基于一般Jacobi多项式J_n~(α,β)(x)(—1<α,β<1)零点的Grnwald插值多项式G_n(f,x);主要证明了G_n(f,x)在(—1,1)内几乎一致收敛于连续函数f(x),并给出了点态逼近估计;拓广和完善了文献[1,2]的结果。     

On summability of weighted Lagrange interpolation. III

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in [15].     

Uniform approximation on [?1, 1] via discrete de la Vallée Poussin means

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss?CJacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.     

Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l ²-norm for some general Mercer kernel matrices are provided. As an example, we give the l ²-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l ²-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms. Supported by the national NSF (No: 10871226) of P.R. China.     

Fourier–Jacobi Type Spherical Functions for Discrete Series Representations of Sp(2, R)

In this paper we define a kind of generalized spherical functions on Sp(2, R). We call it Fourier–Jacobi type, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, R).     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Positivity of the Weights of Interpolatory Quadrature Formulae with Jacobi Abscissae

We consider interpolatory quadrature formulae, relative to the Legendre weight function w(t) = 1 on [–1, 1], having as nodes the zeros of the nth degree Jacobi polynomial P _n ^{(, )} plus the points 1 and –1. We show that in specific domains of and gb the weights of these formulae are almost all positive, exceptions occurring only with the weights corresponding to 1 and –1.     

Conjugate Points on 2-Step Nilpotent Groups

Let N be a simply connected 2-step nilpotent Lie group equipped with a left-invariant metric. We consider the characterizations of Jacobi fields and conjugate points along geodesics emanating from the identity element in N. We obtain a partial result for N and the complete result for N with a one-dimensional center.     

Jacobi transplantation revisited

A transplantation theorem for Jacobi series proved by Muckenhoupt is reinvestigated by means of a suitable variant of Calderón–Zygmund operator theory. An essential novelty of our paper is weak type (1,1) estimate for the Jacobi transplantation operator, located in a fairly general weighted setting. Moreover, L ^p estimates are proved for a class of weights that contains the class admitted in Muckenhoupt’s theorem. Research of ó. Ciaurri and K. Stempak was supported by the grant MTM2006-13000-C03-03 of the DGI. Research of A. Nowak and K. Stempak was supported by MNiSW Grant N201 054 3214285.     

Generalized Convolution for the Discrete Jacobi Transformation

We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences. This revised version was published online in June 2006 with corrections to the Cover Date.     

Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product

where the mass points a_k belong to [−1,1], M_k,i0, i=0,…,N_k−1, and M_k,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants M_k,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in .     

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ? ^N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λ_min. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.     

关于广义Jacobi权的Lebesgue函数的估计

本文证明了由广义Ｊａｃｏｌｂｉ权产生的Ｌｅｂｅｓｇｕｅ函数在（－１,１）中任意固定内闭区间τ上的估计为Ｏ（ｌｎｎ）,这个结果改进了「３」中在一般抽象权时给出的结论     

Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1,1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann–Hilbert (RH) problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the RH problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order 1/n terms in the expansions. A critical step in the analysis of the RH problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order. In addition, the RH approach gives us also strong asymptotics of the orthogonal polynomials near the algebraic singularities in terms of Bessel functions.     

An upper bound on Jacobi polynomials

Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:

where δ_-1<δ₁ are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that

in the region .