Constructive Approximation

, Volume 10, Issue 4, pp 451–468

Regarding thep-norms of radial basis interpolation matrices

  • B. J. C. Baxter
  • N. Sivakumar
  • J. D. Ward
Article

DOI: 10.1007/BF01303522

Cite this article as:
Baxter, B.J.C., Sivakumar, N. & Ward, J.D. Constr. Approx (1994) 10: 451. doi:10.1007/BF01303522

Abstract

A radial basis function approximation has the form
whereϕ:RdR is some given (usually radially symmetric) function, (yj)1n are real coefficients, and the centers (xj)1n are points inRd. For a wide class of functions ϕ, it is known that the interpolation matrixA=(ϕ(xjxk))j,k=1n is invertible. Further, several recent papers have provided upper bounds on ||A−1||2, where the points (xj)1n satisfy the condition ||xjxk||2≥δ,jk, for some positive constant δ. In this paper we calculate similar upper bounds on ||A−1||2 forp≥1 which apply when ϕ decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixAn = (ϕ(jk))j,k=1n when ϕ(x)=(x2+c2)1/2, the Hardy multiquadric. In particular, we show that supn ||An−1|| is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix
enjoys the remarkable property that ||E−1||p = ||E−1||2 for everyp≥1 when ϕ is a Gaussian. Indeed, we also show that this property persists for any function ϕ which is a tensor product of even, absolutely integrable Pólya frequency functions.

AMS classification

41A0541A63

Key words and phrases

Positive definiteRadial functionsPolya frequency functionsToeplitz matricesMultiquadrics

Copyright information

© Springer-Verlag New York Inc 1994

Authors and Affiliations

  • B. J. C. Baxter
    • 1
  • N. Sivakumar
    • 1
  • J. D. Ward
    • 1
  1. 1.Center for Approximation Theory Department of MathematicsTexas A&M UniversityCollege StationU.S.A.
  2. 2.Department of MathematicsUniversity of ManchesterManchesterEngland