Mean square simultaneous approximation of certain classes of functions of two variables by algebraic ``angles''
0000-0003-3016-3575 The paper establishes certain exact inequalities between the best simulta\-neous approximations of functions of two variables and their intermediate derivatives by algebraic ``angles'' and the averaged values of the generalized modulus of continuity of the $r$-th derivatives $\mathcal{D}^{r}f$ $(r\in\mathbb{N})$, where $\mathcal{D}$ is the second-order Chebyshev differential operator in the metric of the space $L_{2,\mu}(Q)$, $Q:=\{(x,y):-1\le x,y\le1\}$ with the Chebyshev weight $\mu:=\mu(x,y)=1/\sqrt{(1-x^2)(1-y^2)}$. The aforementioned generalized modulus of continuity is generated by a special generalized shift operator used for the expansion of an arbitrary function $f\in L_{2,\mu}(Q)$ into a double Fourier–Chebyshev series. The obtained results are presented in the form of inequalities that relate the quantities of the best approximation of functions by algebraic ``angles'' to double integrals involving the modulus of continuity $\Omega_{k}(\mathcal{D}^{r}f;t,\tau)_{2,\mu}$ on the class $L_{2,\mu}^{(r)}:=\bigl\{f\in L_{2,\mu}: \bigl\|\mathcal{D}^{r}f\bigr\|_{2,\mu}<\infty\bigr\}$. These inequalities are exact in the sense that there exists an extremal function $f_{0}\in L_{2,\mu}^{(r)}$, for which the inequalities become equalities.
УДК 517.5
Keywords: approximation by ``angle'', Chebyshev weight, differential operator, Hilbert space, generalized shift operator, modulus of continuity.