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Mathematics of Computation

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Rate of convergence of Shepard's global interpolation formula

Author: Reinhard Farwig
Journal: Math. Comp. 46 (1986), 577-590
MSC: Primary 65D05; Secondary 41A05, 41A25
MathSciNet review: 829627
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Abstract: Given any data points $ {x_1}, \ldots ,{x_n}$ in $ {{\mathbf{R}}^s}$ and values $ S_p^q$ of a function f, Shepard's global interpolation formula reads as follows:

$\displaystyle S_p^0f(x) = \sum\limits_i {f({x_i}){w_i}(x),\quad {w_i}(x) = \vert x - {x_i}{\vert^{ - p}}/\sum\limits_j {\vert x - {x_j}{\vert^{ - p}},} } $

where $ f({x_1}), \ldots ,f({x_n})$ denotes the Euclidean norm in $ \vert \cdot \vert$. This interpolation scheme is stable, but if $ {{\mathbf{R}}^s}$, the gradient of the interpolating function vanishes in all data points. The interpolation operator $ p > 1$ is defined by replacing the values $ S_p^q$ in $ f({x_i})$ by Taylor polynomials of f of degree $ S_p^0f$. In this paper, we investigate the approximating power of $ q \in {\mathbf{N}}$ for all values of p, q and s.

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Keywords: Multivariate interpolation, Shepard's formula, rate of convergence
Article copyright: © Copyright 1986 American Mathematical Society