Orthogonality preserving maps and the Laguerre functional
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- by William R. Allaway PDF
- Proc. Amer. Math. Soc. 100 (1987), 82-86 Request permission
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 $a \ne 0$.References
- W. A. Al-Salam and A. Verma, Some orthogonality preserving operators, Proc. Amer. Math. Soc. 23 (1969), 136–139. MR 249912, DOI 10.1090/S0002-9939-1969-0249912-5
- W. A. Al-Salam and A. Verma, Orthogonality preserving operators, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 6, 833–838. MR 440090
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- Steven M. Roman and Gian-Carlo Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95–188. MR 485417, DOI 10.1016/0001-8708(78)90087-7
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 82-86
- MSC: Primary 33A65; Secondary 42C05
- DOI: https://doi.org/10.1090/S0002-9939-1987-0883405-8
- MathSciNet review: 883405