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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Orthogonality preserving maps and the Laguerre functional


Author: William R. Allaway
Journal: Proc. Amer. Math. Soc. 100 (1987), 82-86
MSC: Primary 33A65; Secondary 42C05
MathSciNet review: 883405
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Abstract: Let $ \Re [x]$ be the usual algebra of all polynomials in the indeterminate $ x$ over the field of real numbers $ \Re $, and let $ \varphi $ be a linear operator mapping $ \Re [x]$ into $ \Re [x]$. In this paper we show that if $ \varphi $ maps every orthogonal polynomial sequence into an orthogonal polynomial sequence, then $ \varphi $ is defined by $ \varphi ({x^n}) = s{(ax + b)^n},n = 0,1,2, \ldots $, where $ s,a,$, and $ b$ belong to $ \Re $, $ s \ne 0$, and $ a \ne 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0883405-8
PII: S 0002-9939(1987)0883405-8
Keywords: Orthogonal polynomials, orthogonality preserving operators, Laguerre functional, Laguerre polynomials, shift operator, scale operator, pseudo-basis
Article copyright: © Copyright 1987 American Mathematical Society