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Orthogonal polynomials on several intervals via a polynomial mapping


Authors: J. S. Geronimo and W. Van Assche
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
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0951620-6
Keywords: Orthogonal polynomials, recurrence coefficients, Jacobi matrices
Article copyright: © Copyright 1988 American Mathematical Society

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