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Mean Convergence of Lagrange Interpolation for Exponential weights on [-1, 1]

Published online by Cambridge University Press:  20 November 2018

D. S. Lubinsky
D. S. Lubinsky*
Affiliation:
Mathematics Department, Witwatersrand University, Wits 2050, South Africa email: 036dsl@cosmos.wits.ac.za
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Abstract

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We obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on [-1, 1], such as

$$w(x)\,=\,\exp \left( -{{\left( 1-{{x}^{2}} \right)}^{-\alpha }} \right),\,\alpha >0$$

or

$$w(x)=\exp \left( -{{\exp }_{k}}{{\left( 1-{{x}^{2}} \right)}^{-\alpha }} \right),k\ge 1,\alpha >0,$$

where ${{\exp }_{k}}=\exp \left( \exp \left( \cdot \cdot \cdot \exp (\,)\cdot \cdot \cdot \right) \right)$ denotes the $k$-th iterated exponential.

Keywords

41A0542C99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 6 , 01 December 1998 , pp. 1273 - 1297
Copyright
Copyright © Canadian Mathematical Society 1998

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