An exponential-trigonometric spline minimizing a seminorm in a Hilbert space

[摘要] In the present paper, using the discrete analogue of the operator$\mathrm{d} ^{6}/\mathrm{d} x^{6}-1$ , we construct an interpolation spline that minimizes the quantity$\int _{0}^{1}(\varphi {'''}(x)+\varphi (x))^{2}\,\mathrm{d}x$ in the Hilbert space$W_{2}^{(3,0)}$ . We obtain explicit formulas for the coefficients of the interpolation spline. The obtained interpolation spline is exact for the exponential-trigonometric functions${{e}^{-x}}$ ,${{e}^{\frac{x}{2}}}\cos ( \frac{\sqrt{3}}{2}x)$ , and${{e}^{\frac{x}{2}}}\sin ( \frac{\sqrt{3}}{2}x )$ .

[学科分类] 航空航天科学

[关键词] Interpolation spline;Hilbert space;The norm minimizing property;Sobolev’s method;Discrete argument function