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Spline approximation and generalized Turán quadratures. (English) Zbl 0857.41008
Summary: In this paper, which is connected with our previous work [the authors, Numerical mathematics, Proc. Int. Conf., Singapore 1988, ISNM, Int. Ser. Numer. Math. 86, 357-365 (1988; Zbl 0659.65010)], we consider the problem of approximating a function \(f\) on the half-line by a spline function of degree \(m\) with \(n\) (variable) knots (multiplicities of the knots are greater or equal than one). In the approximation procedure we use the moments of the function \(r\mapsto f(r)\) and its derivatives at the origin \(r=0\). If the approximation exists, we show that it can be represented in terms of the generalized Turán quadrature relative to a measure depending on \(f\). Also the error in the spline approximation formula is expressed by the error term in the corresponding quadrature formula. A numerical example is included.

41A15 Spline approximation
65D32 Numerical quadrature and cubature formulas
33C65 Appell, Horn and Lauricella functions