Spline approximations to spherically symmetric distributions

Summary We discuss the problem of approximating a functionf of the radial distancer in ^d on 0r< by a spline function of degreem withn (variable) knots. The spline is to be constructed so as to match the first 2n moments off. We show that if a solution exists, it can be obtained from ann-point Gauss-Christoffel quadrature formula relative to an appropriate moment functional or, iff is suitably restricted, relative to a measure, both depending onf. The moment functional and the measure may or may not be positive definite. Pointwise convergence is discussed asn. Examples are given including distributions from statistical mechanics.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561.