Orthogonal Polynomials Associated with Related Measures and Sobolev Orthogonal Polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.     

General Sobolev Orthogonal Polynomials

In this paper, we study orthogonal polynomials with respect to the inner product (f, g)_S^(N) =〈u, fg〉+∑_m=1^N λ_m〈u, f^(m)g^(m) 〉, where λ_m≥0 form=1,…,N, anduis a semiclassical, positive definite linear functional. For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated withu.     

Polynomials with Odd Orthogonal Multiplicity

Let theorthogonal multiplicityof a monic polynomialgover a field be the number of polynomialsfover , coprime togand of degree less than that ofg, such that all the partial quotients of the continued fraction expansion off/gare of degree 1. Polynomials with positive orthogonal multiplicity arise in stream cipher theory, part of cryptography, as the minimal polynomials of the initial segments of sequences which have perfect linear complexity profiles. This paper focuses on polynomials which have odd orthogonal multiplicity; such polynomials are characterized and a lower bound on their orthogonal multiplicity is given. A special case of a conjecture on rational functions over the finite field of two elements with partial quotients of degree 1 or 2 in their continued fraction expansion is also proved.     

Julia Sets of Orthogonal Polynomials

Potential Analysis - For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {Pn} to...     

Finite Differences and Orthogonal Polynomials

By combining finite differences with symmetric functions, we present an elementary demonstration for the limit relation from Laguerre to Hermite polynomials, proposed by Richard Askey. Another limit relation between these two polynomials will also be established.     

Orthogonal Polynomials on a Bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice ? and on the shifted lattice ?+1???. We combine both lattices to obtain the bi-lattice ???(?+1???) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients that satisfy a nonlinear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.     

The L-Appell Symmetric Orthogonal Polynomials

In this paper, we shall be concerned with lowering operators defined on polynomials by means of

$$\begin{aligned} L(x^n)=\mu _nx^{n-1},\ \ n=0,1,\ldots , \ \mu _0=0,\ \ \mu _n\ne 0\ \ (n=1,2,\ldots ). \end{aligned}$$

We determine a necessary and sufficient condition on lowering operators L and a symmetric orthogonal polynomial sets \(\{P_n\}_{n\ge 0}\) such that \(\{P_n\}_{n\ge 0}\) is L-Appell. The resulting polynomials are the generalized Hermite and the symmetric PSs related to Wall and generalized Stieltjes–Wigert. Various properties of the obtained families are singled out: a three-term recurrence relation, explicit expression in term of hypergeometric and basic hypergeometric functions and generating functions.

     

Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.     

Wavelets Based on Orthogonal Polynomials

We present a unified approach for the construction of polynomial wavelets. Our main tool is orthogonal polynomials. With the help of their properties we devise schemes for the construction of time localized polynomial bases on bounded and unbounded subsets of the real line. Several examples illustrate the new approach.

     

Hyperfunctional Weights for Orthogonal Polynomials

The Chebychev polynomials associated to any given moments μ_{n ∞} ⁰ are formally orthogonal with respect to the formal δ-series $$w(x)= {\sum^\infty_0}(- 1)^{n}\mu_{{n}}\delta^{(n)}(x)/n!.$$ We show that this formal weight can be a true hyperfunctional weight if its Fourier transform is a slowly increasing holomorphic function in some tubular neighborhood of the real line. It provides a unifying treatment of real and complex orthogonality of Chebychev polynomials including all classical examples and characterizes Chebychev polynomials having Bessel type orthogonality.     

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

Orthogonal Polynomials on Generalized Julia Sets

We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green’s functions and Parreau–Widom criterion for a special family of real generalized Julia sets.     

Orthogonal Polynomials Satisfying Higher-Order Difference Equations

We introduce a large class of measures with orthogonal polynomials satisfying higher-order difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights of Charlier, Meixner, Krawtchouk, and Hahn by certain variants of the annihilator polynomial of a finite set of numbers.     

Blumenthal''s Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B_n(x)=(x−β_n)B_n−1(x)−α_nxB_n−2(x), with positive recurrence coefficients α_n+1,β_n (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α_n→α and β_n→β and show that the zeros of B_n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.     

Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Laurent 多项式及其零点

本文给出了测度dψ为强分布的一个必要条件,并得到了dψ为强分布时的Laurent多项式最大零点的一个表示。     

Ratio Asymptotics for Orthogonal Matrix Polynomials

Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind.