Abstract
In the error analysis of the process of interpolation by translates of a single basis function, certain spaces of functions arise naturally. These spaces are defined with respect to a seminorm which is given in terms of the Fourier transform of the function. We call this an indirect seminorm. In certain well‐understood cases, the seminorm can be rewritten trivially in terms of the function, rather than its Fourier transform. We call this a direct seminorm. The direct form allows better error estimates to be obtained. In this paper, we shown how to rewrite most of the commonly arising indirect form seminorms in direct form, and begin a little of the analysis required to obtain the improved error estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1978).
J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m-splines, RAIRO Anal. Numer. 12(4) (1978) 325-334.
J. Duchon, Splines minimising rotation-invariant seminorms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, Vol. 571, eds. W. Schempp and K. Zeller (Springer, Berlin, 1977) pp. 85-100.
L. Hörmander, The Analysis of Linear Partial Differential Operators I (Springer, Berlin, 1983).
W.A. Light and H.S.J. Wayne, Some aspects of radial basis function approximation, in: Approximation Theory, Spline Functions and Applications, ed. S.P. Singh (Kluwer Academic, Dordrecht, 1995) pp. 163-190.
W.A. Light and H.S.J. Wayne, Spaces of distributions, interpolation by translates of a basis function, and error estimates, Numer. Math. 81 (1999) 415-450.
W. Madych and S. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988) 77-89.
W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990) 211-230.
W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973).
W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1970).
R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in: Approximation Theory VIII, Vol. I: Approximation and Interpolation, eds. C.K. Chui and L.L. Schumaker (World Scientific, Singapore, 1995) pp. 491-514.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Levesley, J., Light, W. Direct form seminorms arising in the theory of interpolation by translates of a basis function. Advances in Computational Mathematics 11, 161–182 (1999). https://doi.org/10.1023/A:1018971808961
Issue Date:
DOI: https://doi.org/10.1023/A:1018971808961