Regarding thep-norms of radial basis interpolation matrices
- Cite this article as:
- Baxter, B.J.C., Sivakumar, N. & Ward, J.D. Constr. Approx (1994) 10: 451. doi:10.1007/BF01303522
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A radial basis function approximation has the form whereϕ:Rd→R 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(xj−xk))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 ||xj−xk||2≥δ,j≠k, 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(j−k))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.
Key words and phrasesPositive definiteRadial functionsPolya frequency functionsToeplitz matricesMultiquadrics
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