Recursion relation for orthogonal polynomials on the complex plane

Abstract

The article deals with orthogonal polynomials on compact infinite subsets of the complex plane. Orthogonal polynomials are treated as coordinates of generalized eigenvector of a normal operator $A$. It is shown that there exists a recursion that gives the possibility to reconstruct these polynomials. This recursion arises from generalized eigenvalue problem and, actually, this means that every gene alized eigenvector of $A$ is also a generalized eigenvector of $A^*$ with the complex conjugated eigenvalue. If the subset is actually the unit circle, it is shown that the presented algorithm is a generalization of the well-known Szego recursion from OPUC theory.

Key words: Block three-diagonal matrix, orthogonal polynomials, generalized eigenvector, Verblunsky coefficients, Szeg˝o recursion, moment problem.

Full Text

Article Information

Title	Recursion relation for orthogonal polynomials on the complex plane
Source	Methods Funct. Anal. Topology, Vol. 14 (2008), no. 2, 108-116
MathSciNet	MR2432759
zbMATH	1164.42017
Copyright	The Author(s) 2008 (CC BY-SA)

Authors Information

Yu. M. Berezansky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivs'ka, Kyiv, 01601, Ukraine

I. Ya. Ivasiuk
Kyiv National Taras Shevchenko University, Mechanics and Mathematics Faculty, Department of Mathematical Analysis, Kyiv, 01033, Ukraine

O. A. Mokhonko
Kyiv National Taras Shevchenko University, Mechanics and Mathematics Faculty, Department of Mathematical Analysis, Kyiv, 01033, Ukraine 

Citation Example

Yu. M. Berezansky, I. Ya. Ivasiuk, and O. A. Mokhonko, Recursion relation for orthogonal polynomials on the complex plane, Methods Funct. Anal. Topology 14 (2008), no. 2, 108-116.

BibTex

@article {MFAT448,
    AUTHOR = {Berezansky, Yu. M. and Ivasiuk, I. Ya. and Mokhonko, O. A.},
     TITLE = {Recursion relation for orthogonal polynomials on the complex plane},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {14},
      YEAR = {2008},
    NUMBER = {2},
     PAGES = {108-116},
      ISSN = {1029-3531},
  MRNUMBER = {MR2432759},
 ZBLNUMBER = {1164.42017},
       URL = {http://mfat.imath.kiev.ua/article/?id=448},
}