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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Orthogonal polynomials on several intervals via a polynomial mapping
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by J. S. Geronimo and W. Van Assche PDF
Trans. Amer. Math. Soc. 308 (1988), 559-581 Request permission

Abstract:

Starting from a sequence $\{ {p_n}(x; {\mu _0})\}$ of orthogonal polynomials with an orthogonality measure ${\mu _0}$ supported on ${E_0} \subset [ - 1, 1]$, we construct a new sequence $\{ {p_n}(x; \mu )\}$ of orthogonal polynomials on $E = {T^{ - 1}}({E_0})$ ($T$ is a polynomial of degree $N$) with an orthogonality measure $\mu$ that is related to ${\mu _0}$. If ${E_0} = [ - 1, 1]$, then $E = {T^{ - 1}}([ - 1, 1])$ will in general consist of $N$ intervals. We give explicit formulas relating $\{ {p_n}(x; \mu )\}$ and $\{ {p_n}(x; {\mu _0})\}$ and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses $T$ to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 308 (1988), 559-581
  • MSC: Primary 42C05; Secondary 30E05, 33A65
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0951620-6
  • MathSciNet review: 951620