Generalized Trace Formula for Polynomials Orthogonal in Continuous-Discrete Sobolev Spaces

Functional Analysis and Its Applications - In continuous-discrete Sobolev spaces a generalized trace formula for orthogonal polynomials $$\{\widehat{q}_n\}_{n=0}^\infty$$ is obtained. The proof of...     

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.     

圆弧和球面上的广义Ball多项式

本文以Ｂéｚｉｅｒ多项式理论为基础,引进了圆弧上的广义Ｂａｌ曲线及球面三角剖分上的广义Ｂａｌ曲面及其递归算法．     

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.     

Recurrence Formula for Polynomials of Two Variables, Orthogonal with Respect to Rotation Invariant Measures

We give the recurrence formula satisfied by polynomials of two variables, orthogonal with respect to a rotation invariant measure. Moreover, we show that for polynomials satisfying such a recurrence formula, there exists an orthogonality measure which is rotation invariant. We also compute explicitly the recurrence coefficients for the disk polynomials. January 4, 1997. Date revised: October 15, 1997. Date accepted: October 21, 1997.     

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.

     

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.     

Interpolation by Polynomials and Radial Basis Functions on Spheres

The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest. March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999.     

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.     

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.     

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.     

Riesz's Theorem for Orthogonal Matrix Polynomials

We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of V for which the matrix polynomials are dense in the corresponding ² space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set. May 20, 1997. Date revised: January 8, 1998.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

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 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 on Planar Cubic Curves

Foundations of Computational Mathematics - Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic...     

Orthogonal Polynomials on the Sierpinski Gasket

The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket (SG) has given rise to intensive research on analysis on fractals. For instance, a complete theory of polynomials and power series on SG has been developed by one of us and his coauthors. We build on this body of work to construct certain analogs of classical orthogonal polynomials (OP) on SG. In particular, we investigate key properties of these OP on SG, including a three-term recursion formula and the asymptotics of the coefficients appearing in this recursion. Moreover, we develop numerical tools that allow us to graph a number of these OP. Finally, we use these numerical tools to investigate the structure of the zero and the nodal sets of these polynomials.     

A Conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm–Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariñena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux–Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X₂-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X₂-OPSs. The classification includes all cases known to date plus some new examples of X₂-Laguerre and X₂-Jacobi polynomials.