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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Divergence of interpolation polynomials in the complex domain


Author: P. J. O’Hara
Journal: Proc. Amer. Math. Soc. 25 (1970), 690-697
MSC: Primary 30.70
DOI: https://doi.org/10.1090/S0002-9939-1970-0273031-3
MathSciNet review: 0273031
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1918 L. Fejér gave an example of a function $f(z)$, analytic for $|z| < 1$ and continuous for $|z| \leqq 1$, such that the sequence of Lagrange polynomials found by interpolation to $f(z)$ at the roots of unity diverges at a point on the unit circle. More recently S. Ja. Al’per showed that, regardless of how the interpolation points are chosen on the unit circle, a function $g(z)$, analytic for $|z| < 1$ and continuous for $|z| \leqq 1$, exists such that the Lagrange polynomials do not converge uniformly to $g(z)$ for $|z| \leqq 1$. In the present paper we present some theory which sheds some light on the results of Fejér and Al’per. A new example of the divergence of Lagrange polynomials is also presented.


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Keywords: Complex Lagrange polynomials, pointwise convergence, uniform convergence, linear functional, uniform boundedness principle
Article copyright: © Copyright 1970 American Mathematical Society