Spline Approximation and Generalized Turán Quadratures

Milan A. Kova\v cevi\'c and Gradimir V. Milovanovi\'c

Abstract: In this paper, which is connected with our previous work [16], 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.

Keywords: Spline approximation; Turán quadratures; $s$-orthogonal polynomials.

Classification (MSC2000): 41A15, 65D32; 33C45

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