Rate of convergence of Shepard’s global interpolation formula
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- by Reinhard Farwig PDF
- Math. Comp. 46 (1986), 577-590 Request permission
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: \[ S_p^0f(x) = \sum \limits _i {f({x_i}){w_i}(x),\quad {w_i}(x) = |x - {x_i}{|^{ - p}}/\sum \limits _j {|x - {x_j}{|^{ - p}},} } \] where $f({x_1}), \ldots ,f({x_n})$ denotes the Euclidean norm in $| \cdot |$. 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.References
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Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 46 (1986), 577-590
- MSC: Primary 65D05; Secondary 41A05, 41A25
- DOI: https://doi.org/10.1090/S0025-5718-1986-0829627-0
- MathSciNet review: 829627