Moment-preserving spline approximation on finite intervals. (English) Zbl 0644.41005

Frontini, Marco; Gautschi, Walter; Milovanović, Gradimir V.

Numer. Math. 50, 503-518 (1987).

The authors discuss the problem of approximating a function f on the interval [0,1] by a spline function of degree m, with n (variable) knots, matching as many of the initial moments of f as possible. Additional constraints on the derivates of the approximation at one endpoint of [0,1] may also be imposed. They show that, if the approximations exist, they can be represented in terms of generalized Gauss-Lobatto and Gauss- Radau quadrature rules relative to appropriate moment functionals or measures (depending on f). Pointwise convergence as \(n\to \infty\), for fixed \(m>0\), is shown for functions f that are completely monotonic on [0,1], among others. Numerical examples conclude the paper.

Reviewer: A.Bleyer

MSC:

41A15	Spline approximation
65D32	Numerical quadrature and cubature formulas
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

constraints; Gauss-Lobatto; Gauss-Radau quadrature rules; Numerical examples

References:

[1]	Gautschi, W.: A survey of Gauss-Christoffel quadrature formulae. In: E.B. Christoffel, Butzer, P.L., Fehér, F. (eds.), pp. 72–147. Basel: Birkhäuser 1981 · Zbl 0479.65001
[2]	Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput.3, 289–317 (1982) · Zbl 0482.65011 · doi:10.1137/0903018
[3]	Gautschi, W.: An algorithmic implementation of the generalized Christoffel theorem. In: Numerische Integration. Hämmerlin, G. (ed.)., Internat. Ser. Numer. Math., vol. 57, pp. 89–106. Basel: Birkhäuser 1982 · Zbl 0518.65006
[4]	Gautschi, W.: Discrete approximations to spherically symmetric distributions. Numer. Math.44, 53–60 (1984) · Zbl 0508.65012 · doi:10.1007/BF01389754
[5]	Gautschi, W.: Some new applications of orthogonal polynomials. In: Polynômes orthogonaux et applications. Brezinski, C., Draux, A., Magnus, A.P., Maroni, P., Ronveaux, A. (eds.), Lecture Notes Math., vol. 1171, pp. 63–73. Berlin-Heidelberg-New York-Tokyo: Springer 1985 · Zbl 0658.42028
[6]	Gautschi, W., Milovanović, G.V.: Spline approximations to spherically symmetric distributions. Numer. Math.49, 111–121 (1986) · Zbl 0586.41009 · doi:10.1007/BF01389619
[7]	Golub, G.H., Kautsky, J.: Calculation of Gauss quadratures with multiple free and fixed knots. Numer. Math.41, 147–163 (1983) · Zbl 0525.65010 · doi:10.1007/BF01390210
[8]	Widder, D.V.: The Laplace Transform. Princeton: University Press 1941 · Zbl 0063.08245