The constant factor in error estimates of the variational-difference approximation

S. G. Mikhlin

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 80 (1978), pp. 125-166

Résumé

Let $U\in W_p^{(2s)}(0,1)$ and let the original functions $\omega_{q,s}(x)$, $0\leqslant q\leqslant s-1$ vanish outside the interval $[0,2]$, while on each of the intervals $(0,1)$ and $(1,2)$ they are polynomials of degree $2s-1$. Let \begin{equation} U^h(x)=\sum_{q=0}^{2s-1}\sum_{j=-1}^{2n-1}h^2U^{(q)}((j+1)h)\omega_{q,s} \biggl(\dfrac{x}{h}-j\biggr),\quad h=\dfrac{1}{2n}. \tag{1} \end{equation} Then, as we know, \begin{equation} \|U-U^h\|_{l_p(\overline{s})}\leqslant C(s,\overline{s})h^{2s-\overline{s}}\|U^{(2s)}\|_{L_p(0,1)}\quad \overline{s}\leqslant s; \tag{2} \end{equation} similar results were also obtained for functions of many variables. In this article we derive bounds on the polynomials $\sigma_{q,s}(t)=\omega_{q,s}(t+1)$, $0\leqslant t\leqslant1$ and their derivatives of order $\leqslant s$ in the metrics $C$ and $L_p$; our bounds prove to be essentially better than Markovian. A bound on $C(s,\overline{s})$ in inequality (2) is obtained. In the many-variable case we consider the approximation of functions from the classes $C(\Omega)$ and $W_p^{(2s)}(\Omega)$ by functions $U^t$ analogous to the functions (1); the original functions are obtained by multiplying one-dimensional piecewise-polynomial original functions. For the functions of the class $W_p^{(2s)}(\Omega)$ the corresponding constant $C(s,\overline{s})$ depends on two additional quantities, which are called here the averaging constant and the extension constant. An estimate of the “averaging constant” is obtained; the “extension constant” is estimated for the Hestenes extension.