Numerical Algorithms 10(1995)401-419 401 Convolution kernels based on thin-plate splines J. Levesley Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, UK E-mail: jll @mcs.le.ac.uk Received 10 January 1995 Communicated by P.J. Laurent Quasi-interpolation using radial basis functions has become a popular method for con- structing approximations to continuous functions in many space dimensions. In this paper we discuss a procedure for generating kernels for quasi-interpolation, using functions which have series expansions involving terms like r ~ logr. It is shown that such functions are suitable if and only if c~ is a positive even integer and the spatial dimension is also even. Keywords: Radial basis functions, thin-plate splines, convolution kernels. AMS(MOS) subject classification: 41A30, 41A63. 1. Introduction In recent years the approximation of continuous functions f : R" ~ R, using radial basis functions, has received much attention. Given a function ~b : Ii~+ ~ IlL a radial function is one of the form q~ o [] II, where [[ [I is the Euclidean distance. Then, we seek approximations to f of the form f ~ Z (llx - ay[I), x c R", j~J where {a/}/~s c_ ~" and J is a (possibly infinite) index set. To date, two main procedures have been used for producing approximations; interpolation and quasi-interpolation. For information on the former see Powell [8] and Buhmann and Powell [2]. In this paper we concentrate on quasi-interpola- tion. For this procedure we require a kernel G which satisfies the two following properties. 9 G is integrable on II~". 9 fR, G=I. We use G to construct a sequence of approximations to the Dirac distribution 6, which has the property f(x) = JR[ f(Y)6(x - y) dy. 9 J,C. Baltzer AG, Science Publishers