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 $$lambda_N=max_{0le j le N-1} (eta_{j+1}-eta_{j})$$ ....     

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...     

Estimate of the Lebesgue Function of Fourier Sums in Terms of Modified Meixner Polynomials

Mathematical Notes - The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials m (x), n = 0,1,..., which generate, for α...     

按有理函数系展开函数成Fourier级数

In this paper we give some orthogonal system of rational functions on the unit circumference; We obtained the Riemann Lemma and some tests of convergence of Fourier series for this orthogonal system with the poles {α_k}₀^∞ satisfying some conditions, The conditions we obtained are weaker than those established in [2].     

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.     

广义对角多项式指数和的估计

利用高斯和与次数矩阵Smith标准形的不变因子,给出了有限域上广义对角多项式指数和的估计,从而改进了Deligne-Weil型估计这类多项式指数和的结果.     

Three-Term Recurrence Relation for Polynomials Orthogonal with Respect to Harmonic Measure

We prove that a three-term recurrence relation for analytic polynomials orthogonal with respect to harmonic measure in a simply connected domain G exists if and only if G is an ellipse.     

Some Orthogonal Polynomials Related to Elliptic Functions

We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.     

General Lebesgue Constants for Linear Mean Subsequences of Fourier Sums

Simple sufficient conditions for the boundedness of the norms of the general matrix operators constructed from the subsequences of partial sums of trigonometric Fourier series with polynomial order of growth are obtained. They guarantee a particular rate of approximation for subclasses of continuous functions.     

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

We consider the orthogonal polynomials with respect to the measure over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N . The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g‐function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of K ^c with a pole at ∞ . This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the n^th orthogonal polynomial times the orthogonality measure, i.e., The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter in a double scaling limit near the critical point given by we observe the Hastings‐McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.     

有限域上多项式指数和的p-adic估计

设f为有限域上的一个多元多项式.它的次数矩阵Df由出现在f中的次数向量构成.本文利用Df给出了多项式f的指数和的一个p-adic估计,它改进和推广了该方向许多已知的结果.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.     

Recurrence Relations for the Coefficients in Series Expansions with Respect to Semi-Classical Orthogonal Polynomials

Let {P _k } be a sequence of the semi-classical orthogonal polynomials. Given a function f satisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients a _k [f] in f= _k a _k [f]P _k . A systematic use of basic properties (including some nonstandard ones) of the polynomials {P _k } results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

Estimates for Commutators of Orthogonal Fourier Series

In this paper we study weighted norm inequalities for the commutators[b, S_n] where b is a BMO function and S_n denotes the nth partialsum of the Fourier series relative to a system of orthogonalpolynomials on [–1, 1] with respect to general weights.Results about generalized Jacobi and Bessel Fourier series areobtained.     

Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces. J.M. Rodriguez supported in part by three grants from M.E.C. (MTM 2006-13000-C03-02, MTM 2006-11976 and MTM 2007-30904-E), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain. J.M. Sigarreta supported in part by a grant from M.E.C. (MTM 2006-13000-C03-02), Spain, and by a grant from U.C.III M./C.A.M. (CCG07-UC3M/ESP-3339), Spain.