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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The sharpness of Lorentz's theorem on incomplete polynomials


Authors: E. B. Saff and R. S. Varga
Journal: Trans. Amer. Math. Soc. 249 (1979), 163-186
MSC: Primary 41A25; Secondary 33A65, 41A60
MathSciNet review: 526316
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Abstract: For any fixed $ \theta $ with $ 0 < \theta < 1$, G. G. Lorentz recently showed that bounded sequences $ \{\Sigma_{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }$ of incomplete polynomials on $ [ - 1, + 1]$ tend uniformly to zero on closed intervals of $ [ - 1,\Delta (\theta ))$, where $ 2{\theta ^2} - 1 \leqslant \Delta (\theta ) < 2\theta - 1$. In this paper, we show that $ \Delta (\theta ) = 2{\theta ^2} - 1$ is best possible, and that the geometric convergence to zero of such sequences on closed intervals $ [{t_0},{t_1}]$ can be precisely bounded above as a function of $ {t_j}$ and $ \theta $. Extensions of these results to the complex plane are also included.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0526316-X
PII: S 0002-9947(1979)0526316-X
Keywords: Incomplete polynomials, uniform convergence, geometric convergence, Jacobi polynomials, method of steepest descents
Article copyright: © Copyright 1979 American Mathematical Society