Radial basis function approximations as smoothing splines

Radial basis function methods for interpolation can be interpreted as rough ness-minimizing splines. Although this relationship has already been establ ished for radial basis functions of the form g(r) = r(alpha) and g(r) = r(a lpha) log(r), it is extended here to include a much larger class of functio ns. This class includes the multiquadric g(r) = (r(2) + c(2))(1/2) and inve rse multiquadric g(r)=(r(2) + c(2))(-1/2) functions as well as the Gaussian exp(-r(2)/D). The crucial condition is that the Fourier transform of g(/x/ ) be positive, except possibly at the origin. The appropriate measure of ro ughness is defined in terms of this Fourier transform. To allow for possibi lity of noisy data, the analysis is presented within the general framework of smoothing splines, of which interpolation is a special case. Two diagnos tic quantities, the cross-validation function and the sensitivity, indicate the accuracy of the approximation. (C) 1999 Elsevier Science Inc. ALI Figh ts reserved.