We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.

We prove that the error of the best nonlinear L p -approximation by piecewise constants on convex partitions is O N − 2 d+1 , where N is the number of cells, for all functions in the Sobolev space W 2 q (Ω) on a cube Ω ⊂ R d , d 2, as soon asis achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev-Slobodeckij spaces W r q (Ω) embedded in L p (Ω), some of which also improve the standard estimate O N − 1 d known to be optimal on isotropic partitions.

No abstract

УДК 517.5Отримано деякі нові нерівності типу Колмогорова, що оцінюють норму дробової похідної в сенсі Маршо функції визначеної на додатній півосі, через і Також розв'язано споріднені задачі: задачу Стєчкіна про найкраще наближення оператора лінійними обмеженими операторами і задачу про найкраще відновлення оператора на класі, елементи якого відомі з похибкою.

We consider the problem of finding sharp inequalities for the norms of derivatives of the functions. This classical problem arises in Approximation Theory in the beginning of XX century in works of E. Landau, J. Hadamard, G.H. Hardy and J.E. Littlewood. A thorough overview of many known results and related problems can be found in surveys [1, 2] and the book [3].Recall that $L_{2,r;\alpha,\beta}((-1,1))$, $r\in \mathbb{N}$ and $\alpha,\beta> -1$, is the space of measurable functions $x:(-1,1)\to\mathbb{R}$ such that $\|x\|_{2,r;\alpha,\beta} := \int_{-1}^{1} |x(t)|^2(1-t)^{\alpha+r}(1+t)^{\beta+r}\,{\rm d}t < \infty$, and $L_{2,e^{-t^2}}(\mathbb{R})$ is the space of measurable functions $x:\mathbb{R}\to\mathbb{R}$ such that $\|x\|_{2,e^{-t^2}} := \int_{-\infty}^{+\infty} |x(t)|^2e^{-t^2}\,{\rm d}t < \infty$. S.Z. Rafalson [7], S.Z. Rafalson and I.V. Berdnikova [5] obtained analogues of Hardy-Littlewood-Polya inequalities for the norms of derivatives of functions in spaces $L_{2,r;\alpha,\beta}((-1,1))$ and $L_{2,e^{-t^2}}(\mathbb{R})$. Namely, they established sharp inequalities that estimate $\left\|x^{(k)}\right\|_{2,k;\alpha\beta}$, $k\in\mathbb{N}$ and $0 < k < r$, in terms of $\|x\|_{2,0;\alpha,\beta}$ and $\left\|x^{(r)}\right\|_{2,r;\alpha,\beta}$, and sharp inequalities that estimate $\left\|x^{(k)}\right\|_{2,e^{-t^2}}$ in terms of $\left\|x\right\|_{2,e^{-t^2}}$ and $\left\|x^{(r)}\right\|_{2,e^{-t^2}}$. In this paper we obtain the analogues of Taikov-Shadrin inequalities for the norms of derivatives in spaces $L_{2,r;\alpha,\beta}((-1,1))$ and $L_{2,e^{-t^2}}(\mathbb{R})$. Namely, we obtain sharp inequalities that estimate $\left|x^{(k)}(t_0)\right|$, $t_0\in(-1,1)$, $k\in\mathbb{Z}_+$ and $k < r$, in terms of $\|x\|_{2,0;\alpha,\beta}$ and $\left\|x^{(r)}\right\|_{2,r;\alpha,\beta}$, and sharp inequalities that estimate $\left|x^{(k)}(t_0)\right|$, $t_0\in \mathbb{R}$, $k\in\mathbb{Z}_+$ and $k < r$, in terms of $\|x\|_{2,e^{-t^2}}$ and $\left\|x^{(r)}\right\|_{2,e^{-t^2}}$.