Exact bounds for simultaneous approximation of functions of two variables and their derivatives by bilinear splines

M. Sh. Shabozov

Matematičeskie zametki, Tome 59 (1996) no. 1, pp. 142-152

Résumé

We find the exact value of the expression \begin{multline}\quad \varepsilon^{(l,q)}\bigl(W^{(r,s)}H^{\omega_1,\omega_2}(G)\bigr)=\sup\bigl\{\|f^{(l,q)}(\cdot,\cdot) -S_{1,1}^{(l,q)}(f;\cdot,\cdot)\|_{C(G)}: f\in W^{(r,s)}H^{\omega_1,\omega_2}(G)\bigr\}, \end{multline} where $\varphi^{(l,q)}(x,y)=\partial^{1+q}\varphi/\partial x^l\partial y^q$ ($l,q=0,1$, $1\le l+q\le2$) and $S_{1,1}(f;x,y)$ is a bilinear spline interpolating $f(x,y)$ in the nodes of the grid $\Delta_{mn}=\Delta_m^x\times\Delta_n^y$ with $\Delta_m^x$: $x_i=i/m$ ($i=\overline{0,m}$), $\Delta_n^y$: $y_j=j/n$ ($j=\overline{0,n}$). Here $W^{(r,s)}H^{\omega_1,\omega_2}(G)$ is the class of functions $f(x,y)$ with continuous derivatives $f^{(r,s)}(x,y)$ ($r,s=0,1$, $1\le r+s\le2$) on the square $G=[0,1]\times[0,1]$ and with the modulus of continuity satisfying the inequality ($\omega(f^{(r,s)};t,\tau)\le\omega_1(t)+\omega_2(\tau)$, where $\omega_1(t)$ and $\omega_2(t)$ are the given moduli of continuity.