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Spline approximations to spherically symmetric distributions. (English) Zbl 0586.41009
We discuss the problem of approximating a function f of the radial distance r in \({\mathbb{R}}^ d\) on \(0\leq r<\infty\) by a spline function of degree m with n (variable) knots. The spline is to be constructed so as to match the first 2n moments of f. We show that if a solution exists, it can be obtained from an n-point Gauss-Christoffel quadrature formula relative to an appropriate moment functional or, if f is suitably restricted, relative to a measure, both depending on f. The moment functional and the measure may or may not be positive definite. Pointwise convergence is discussed as \(n\to \infty\). Examples are given including distributions from statistical mechanics.

41A15 Spline approximation
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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