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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.

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