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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.

MSC:
65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
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