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A Bernstein-type inequality for the Jacobi polynomial


Authors: Yunshyong Chow, L. Gatteschi and R. Wong
Journal: Proc. Amer. Math. Soc. 121 (1994), 703-709
MSC: Primary 33C45
DOI: https://doi.org/10.1090/S0002-9939-1994-1209419-X
MathSciNet review: 1209419
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Abstract: Let $ P_n^{(\alpha ,\beta )}(x)$ be the Jacobi polynomial of degree n. For $ - \frac{1}{2} \leq \alpha ,\beta \leq \frac{1}{2}$, and $ 0 \leq \theta \leq \pi $, it is proved that

$\displaystyle {(\sin \frac{\theta }{2})^{\alpha + \frac{1}{2}}}{(\cos \frac{\th... ...array}{*{20}{c}} {n + q} \\ n \\ \end{array} } \right){N^{ - q - \frac{1}{2}}},$

where $ q = \max (\alpha ,\beta )$ and $ N = n + \frac{1}{2}(\alpha + \beta + 1)$. When $ \alpha = \beta = 0$, this reduces to a sharpened form of the well-known Bernstein inequality for the Legendre polynomial.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1209419-X
Keywords: Jacobi polynomial, Bernstein inequality, hypergeometric function
Article copyright: © Copyright 1994 American Mathematical Society