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

Abstract

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

Keywords

• Interpolation spline
• Hilbert space
• The norm minimizing property
• Sobolev’s method
• Discrete argument function