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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