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

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 ${{e}^{\frac{x}{2}}}\sin ( \frac{\sqrt{3}}{2}x )$ .