Constrained energy problems with applications to orthogonal polynomials of a discrete variable
type: Item
instance of: scholarly article
title: Constrained energy problems with applications to orthogonal polynomials of a discrete variable
zbMATH Open document ID: 0898.31003
DOI: 10.1007/BF02843160
published in: Journal d'Analyse Mathématique
publication date: 1998-11-02T00:00:00Z
review text: Since the pioneering works of Mhaskar, Saff, Gonchar and Rakhmanov in the eighties, it is known that the equilibrium distribution for the logarithmic potential in presence of an external field is an essential tool in the study of the asymptotic properties of sequences of orthogonal polynomials with respect to a varying weight. On the other hand, recently Rakhmanov has studied the solution of the classical logarithmic energy minimization problem, but considered in the subclass of the unit measures bounded from above by a fixed measure. He has also shown its connection with the limit zero distribution of (diagonal) sequences of orthogonal polynomials of a discrete variable. The authors combine both approaches to the problem of minimizing the logarithmic energy in presence of an external field in the class of unit measures subject to constraints as above. In particular, they extend to this case the classical results about existence, uniqueness, characterization and a representation of the minimal measure, as well as a saturation property. As expected, the solution describes the zero asymptotics of ray sequences of orthogonal polynomials of a discrete variable. Explicit results are obtained for Krawtchouk polynomials.
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1146580
zbMATH Keywords: logarithmic potential
zbMATH Keywords: external field
zbMATH Keywords: constrained problem
zbMATH Keywords: equilibrium distribution
zbMATH Keywords: orthogonal polynomials of a discrete variable
zbMATH Keywords: logarithmic energy minimization problem
author: P. D. Dragnev
author: E. B. Saff
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1146580