Spline approximations to spherically symmetric distributions

Abstract

We discuss the problem of approximating a functionf of the radial distancer in ℝ^d on 0≦r<∞ 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.