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Interpolation splines minimizing a semi-norm

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Abstract

Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in \(K_2(P_2)\), where \(K_2(P_2)\) is the space of functions \(\phi \) such that \(\phi ^{\prime } \) is absolutely continuous, \(\phi ^{\prime \prime } \) belongs to \(L_2(0,1)\) and \(\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty \). Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the trigonometric functions \(\sin x\) and \(\cos x\). Finally, in a few numerical examples the qualities of the defined splines and \(D^2\)-splines are compared. Furthermore, the relationship of the defined splines with an optimal quadrature formula is shown.

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Acknowledgments

We are very grateful to the reviewer for remarks and suggestions, which have improved the quality of the paper.

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Correspondence to Gradimir V. Milovanović.

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Dedicated to Claude Brezinski and Sebastiano Seatzu on their 70th birthday.

The work of the second author was supported in part by the Serbian Ministry of Education, Science and Technological Development.

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Hayotov, A.R., Milovanović, G.V. & Shadimetov, K.M. Interpolation splines minimizing a semi-norm. Calcolo 51, 245–260 (2014). https://doi.org/10.1007/s10092-013-0080-x

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