Orthogonal polynomials and quadratic extremal problems

Author:
J. M. McDougall

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2341-2357

MSC (2000):
Primary 30A10, 31C25; Secondary 30D55, 33C45, 49J50

DOI:
https://doi.org/10.1090/S0002-9947-02-02960-4

Published electronically:
February 1, 2002

MathSciNet review:
1885655

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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.

**1.**T.S. Chihara,*An Introduction to Orthogonal Polynomials*, Gordon and Breach, 1978. MR**58:1979****2.**D. Dickinson,*On Lommel and Bessel Polynomials*, Proc. Amer. Math. Soc.,**5**, 1954, pp. 946-956. MR**19:263c****3.**J. Dombrowski,*Spectral Properties of Real Parts of Shift Operators*, Indiana Univ. Math. J.,**29**, No. 2, 1980, pp. 249-259. MR**81b:47039****4.**Bateman Manuscript Project,*Higher Transcendental Functions*, McGraw-Hill Book Company, N.Y., Vols. 1,2, 1953, Vol. 3, 1955. MR**15:419i**; MR**16:586c****5.**S.D. Fisher,*A Quadratic Extremal Problem on the Dirichlet Space*, Complex Variables Theory Appl., 1995,**26**, pp. 367-380. MR**96i:30030****6.**I.C. Gohberg and M.G. Krein,*Introduction to the Theory of Linear Non-Self Adjoint Operators*, Transl. Math. Monos., vol. 18, Amer. Math. Soc., 1969. MR**39:7447****7.**W. Rudin,*Functional Analysis*, second edition, McGraw-Hill, Inc., 1991. MR**92k:46001****8.**M.H. Stone,*Linear Transformations in Hilbert Space*, Amer. Math. Soc., New York, 1932. (Reprinted 1990, MR**99k:47001**)**9.**G.N. Watson,*A Treatise on the Theory of Bessel Functions,*2nd edition, Cambridge University Press, 1966. (Reprinted 1995, MR**96i:33010**)

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Additional Information

**J. M. McDougall**

Affiliation:
Department of Mathematics and Computer Science, Colorado College, Colorado Springs, Colorado 80903

Email:
JMcDougall@ColoradoCollege.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02960-4

Received by editor(s):
July 7, 1998

Received by editor(s) in revised form:
May 8, 2001

Published electronically:
February 1, 2002

Article copyright:
© Copyright 2002
American Mathematical Society