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Interpolation in the limit of increasingly flat radial basis functions. (English) Zbl 1006.65013
The paper is concerned with the interpolation by radial basis functions (RBFs) containing a free parameter \(\varepsilon\).
If this parameter tends to zero, the RBF becomes increasingly flat and the linear system which must be solved for the interpolation becomes very ill-conditioned. The authors prove that under mild assumptions the RBF interpolant \(s(x,\varepsilon)\) associated with the RBF \[ \varphi(x)= a_0+ a_1(\varepsilon r)^2+ a_2(\varepsilon r)^4+\cdots \] satisfies \(\lim_{\varepsilon\to 0} s(x,\varepsilon)= L_N(x)\), where \(L_N(x)\) is the Lagrange interpolating polynomial for \(f\).
The authors sketch some preliminary observations regarding the limit in the 2D case.

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
Full Text: DOI
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