A scheme for interpolation by Hankel translates of a basis function

Golomb and Weinberger [1] described a variational approach to interpolation which reduced the problem to minimizing a norm in a reproducing kernel Hilbert space generated by means of a small number of data points. Later, Duchon [2] defined radial basis function interpolants as functions which minimize a suitable seminorm given by a weight in spaces of distributions closely related to Sobolev spaces. These minimal interpolants could be written as a linear combination of translates of a single function φ, the so-called basis function, plus a polynomial. Light and Wayne [3] extended Duchon’s class of weight functions, which in turn allowed for non-radial basis functions in their scheme. Following the approach of Light and Wayne, we discuss interpolation of complex-valued functions defined on the positive real axis I by certain spaces of Sobolev type involving the Hankel transformation and powers of the Bessel operator. The set of interpolation points will be a subset {a1, . . . , an} of I and the interpolants will take the form