Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] T. Bloom, Kergin interpolation of entire functions on $ {C^n}$, Duke Math. J. 48 (1981), 69-83. MR 610176 (83k:32005)
  • [2] A. S. Cavaretta, Jr., C. A. Micchelli and A. Sharma, Multivariate interpolation and the Radon transform, Math. Z. 174 (1980), 263-279. MR 593824 (81m:41036)
  • [3] A. O. Gelfond, Calcul des differences finis, Dunod, Paris, 1963. MR 0157139 (28:376)
  • [4] P. Kergin, A natural interpolation of $ {C^k}$ functions, J. Approx. Theory 29 (1980), 278-293. MR 598722 (82b:41007)
  • [5] C. A. Micchelli, A constructive approach to Kergin interpolation in $ {{\mathbf{R}}^k}$: Multivariate $ B$-splines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), 485-497. MR 590212 (84i:41002)
  • [6] E. Stein, Boundary values of holomorphic functions of several complex variables, Math. Notes, Princeton Univ. Press, Princeton, N.J.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A05, 30E05

Retrieve articles in all journals with MSC: 41A05, 30E05

Additional Information

Keywords: Kergin interpolation, multivariate approximation, divided difference
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society