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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
##### Keywords:
Lagrange polynomials; interpolation; radial basis functions
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##### References:
 [1] Buhmann, M.; Dyn, N., Spectral convergence of multiquadric interpolation, Proc. Edinburgh math. soc., 36, 2, 319-333, (1993) · Zbl 0791.41002 [2] Madych, W.R.; Nelson, S.A., Error bounds for multiquadric interpolation, (), 413-416, College Station, TX, 1989 · Zbl 0738.41007 [3] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions. II, Math. comp., 54, 211-230, (1990) · Zbl 0859.41004 [4] Carlson, R.E.; Foley, T.A., The parameter R2 in multiquadric interpolation, Computers math. applic., 21, 9, 29-42, (1991) · Zbl 0725.65009 [5] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. comp. math., 11, 193-210, (1999) · Zbl 0943.65017 [6] Riemenschneider, S.D.; Sivakumar, N., Gaussian radial-basis functions: cardinal interpolation of ℓ^{p} and power-growth data, Adv. comp. math., 11, 229-251, (1999) · Zbl 0939.41002 [7] Baxter, B.J.C., The asymptotic cardinal function of the multiquadratic φ(r) = (r2 + c2)$$12$$ as c → ∞, Computers math. applic., 24, 12, 1-6, (1992), Special Issue: “Advances in the Theory and Applications of Radial Basis Functions” · Zbl 0764.41016 [8] B. Fornberg, T.A. Driscoll, G. Wright and R. Charles, Observations on the behavior of radial basis functions near boundaries, Computers Math. Applic., (this issue). · Zbl 0999.65005
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