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

 

Convergence of multivariate polynomials interpolating on a triangular array


Authors: T. N. T. Goodman and A. Sharma
Journal: Trans. Amer. Math. Soc. 285 (1984), 141-157
MSC: Primary 41A05; Secondary 30E05
MathSciNet review: 748835
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Abstract: Given a triangular array of complex numbers, it is well known that for any function $ f$ smooth enough, there is a unique polynomial $ {G_n}f$ of degree $ \leq n$ such that on each of the first $ n + 1$ rows of the array the divided difference of $ {G_n}f$ coincides with that of $ f$. This result has recently been generalized to give a unique polynomial $ {\mathcal{G}_n}f$ in $ k$ variables $ (k > 1)$ of total degree $ \leq n$ which interpolates a given function $ f$ on a triangular array in $ {C^k}$. In this paper we extend some results of A. O. Gelfond and derive formulas for $ {\mathcal{G}_n}f$ and $ f - {\mathcal{G}_n}f$ to prove some results on convergence of $ {\mathcal{G}_n}f$ to $ f$ as $ n \to \infty $ under various conditions on $ f$ and on the triangular array.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0748835-5
PII: S 0002-9947(1984)0748835-5
Keywords: Kergin interpolation, multivariate approximation, divided difference
Article copyright: © Copyright 1984 American Mathematical Society