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Transactions of the American Mathematical Society

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

 
 

 

Summability of Jacobi series


Author: Richard Askey
Journal: Trans. Amer. Math. Soc. 179 (1973), 71-84
MSC: Primary 42A56; Secondary 33A65
DOI: https://doi.org/10.1090/S0002-9947-1973-0315351-7
MathSciNet review: 0315351
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Abstract: The positivity of some Cesàro mean is proven for Jacobi series $ \Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac{1}{2}$. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the $ (C,\alpha + \beta + 2)$ means is conjectured and proven for some $ (\alpha ,\beta )$. One consequence of this conjecture would be the complete monotonicity of $ {x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1$.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0315351-7
Keywords: Positive Cesàro sum, Jacobi polynomial, Bessel function, Poisson kernel, delayed means, Lagrange interpolation
Article copyright: © Copyright 1973 American Mathematical Society

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