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