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Baxter, Brad J.C. (2008)
Publisher: Springer
Languages: English
Types: Article
Subjects: ems
A radial basis function (RBF) has the general form\ud $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration.
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    • [1] B. J. C. Baxter (1994), \Norm estimates for inverses of Toeplitz distance matrices", J. Approx. Theory 79:222{242.
    • [2] B. J. C. Baxter and A. Iserles (2003), \On the foundations of computational mathematics", in Handbook of Numerical Analysis XI (P.G. Ciarlet and F. Cucker, eds), North-Holland, Amsterdam, 3{34.
    • [3] M. D. Buhmann (2003), Radial Basis Functions: Theory and Implementations, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
    • [4] W. F. Donoghue, Jr., Distributions and Fourier Transforms, Academic Press, New York.
    • [5] A. Edelman and N. Raj Rao (2005), \Random matrix theory", Acta Numerica: 14: 233{297.
    • [6] G. Fasshauer (2007), Meshfree Approximation Methods with Matlab, World Scienti c Publishing.
    • [7] F. G. Friedlander and M. Joshi (1999), Introduction to the Theory of Distributions, Cambridge University Press.
    • [8] I. R. H. Jackson (1988), \Convergence Properties of Radial Basis Functions", Constr. Approx. 4: 243{264.
    • [9] F. John (1955), Plane waves and spherical means applied to partial di erential equations, Interscience, New York.
    • [10] D. S. Jones (1982), The Theory of Generalised Functions, Cambridge University Press.
    • [11] W. A. Light and E. W. Cheney (2000), A Course in Approximation Theory, Brooks/Cole.
    • [12] W. R. Madych and S. A. Nelson (1990), \Polyharmonic cardinal splines", J. Approx. Theory 60: 141{156.
    • [13] C. A. Micchelli (1986), \Interpolation of scattered data: distance matrices and conditionally positive functions", Constr. Approx. 2: 11-22.
    • [14] V. Milman and G. Schechtman (1986), Asymptotic theory of nite dimensional normed spaces, Lecture Notes in Mathematics 1200, Springer.
    • [15] T. Poggio and S. Smale (2003), \The Mathematics of Learning: Dealing with Data", Notices of the AMS, May 2003: 537{544.
    • [16] M. J. D. Powell (1992), \The Theory of Radial Basis Functions in 1990", in Advances in Numerical Analysis, vol. II, ed. W. A. Light, Oxford University Press, pp. 105{210.
    • [17] W. Rudin (1991), Functional Analysis, McGraw-Hill.
    • [18] I. J. Schoenberg (1938), \Metric spaces and completely monotone functions", Ann. of Math. 39: 811-841.
    • [19] H. Wendland (2004), Scattered Data Approximation, Cambridge University Press.
    • [20] E. T. Whittaker and G. N. Watson (1927), A Course of Modern Analysis, Cambridge University Press.
    • [21] D. V. Widder (1946), The Laplace Transform, Princeton University Press. School of Economics, Mathematics and Statistics,, Birkbeck College, University of London,, Malet Street, London WC1E 7HX, England E-mail address:
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