Multivariate interpolation and conditionally positive definite functions. II

Madych, W. R.; Nelson, S. A.

doi:10.1090/S0025-5718-1990-0993931-7

Multivariate interpolation and conditionally positive definite functions. II
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by W. R. Madych and S. A. Nelson PDF

Math. Comp. 54 (1990), 211-230 Request permission

Abstract:

We continue an earlier study of certain spaces that provide a variational framework for multivariate interpolation. Using the Fourier transform to analyze these spaces, we obtain error estimates of arbitrarily high order for a class of interpolation methods that includes multiquadrics.

References

J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The theory of splines and their applications, Academic Press, New York-London, 1967. MR 0239327
N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475 (English, with French summary). MR 143935
A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., Vol. IV, American Mathematical Society, Providence, R.I., 1961, pp. 33–49. MR 0143037

Distributions and Fourier transforms

Jean Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, pp. 85–100. MR 0493110
Jean Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^{m}$-splines, RAIRO Anal. Numér. 12 (1978), no. 4, 325–334, vi (French, with English summary). MR 519016, DOI 10.1051/m2an/1978120403251
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
A. Gorny, Contribution à l’étude des fonctions dérivables d’une variable réelle, Acta Math. 71 (1939), 317–358 (French). MR 848, DOI 10.1007/BF02547758
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578
W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988), no. 4, 77–89. MR 986343
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686