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Proceedings of the American Mathematical Society

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

 

 

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
MathSciNet review: 0273031
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Abstract: In 1918 L. Fejér gave an example of a function $ f(z)$, analytic for $ \vert z\vert < 1$ and continuous for $ \vert z\vert \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 $ \vert z\vert < 1$ and continuous for $ \vert z\vert \leqq 1$, exists such that the Lagrange polynomials do not converge uniformly to $ g(z)$ for $ \vert z\vert \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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DOI: https://doi.org/10.1090/S0002-9939-1970-0273031-3
Keywords: Complex Lagrange polynomials, pointwise convergence, uniform convergence, linear functional, uniform boundedness principle
Article copyright: © Copyright 1970 American Mathematical Society