Interpolation splines minimizing a semi-norm |
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Authors: | Abdullo R Hayotov Gradimir V Milovanovi? Kholmat M Shadimetov |
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Affiliation: | 1. Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan 2. Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia
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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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