On optimization of cubature formulae for Sobolev classes of functions defined on star domains

Abstract

We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm of their distributional gradient. Thus we generalize the known solution of this optimization problem in the case, when the domain of the functions is convex. Let $Q\subset \mathbb{R}^d$, $d\in\mathbb{N}$, be a nonempty bounded open set. By $W^{1,p}(Q)$, $p\in [1,\infty]$, we denote the Sobolev space of functions $f\colon Q\to \mathbb{R}$ such that $f$ and all their (distributional) partial derivatives of the first order belong to $L_p(Q)$. For $x=(x^1,\dots, x^d)\in \mathbb{R}^d$ and $q\in [1,\infty)$ we set$|x|_q:= \Big(\sum_{k=1}^d|x^k|^q\Big)^\frac {1}{q},$ $|x|_\infty:= \max\{|x^k|\colon k\in\{1,\ldots, d\}\}$, and $W^{\infty}_{p}(Q):=\{f\in W^{1,p}(Q)\colon \|\,|\nabla f|_1\,\|_{L_p(Q)}\leq 1\},$ where $\nabla f=(\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_d})$, $p\in[1,\infty]$. In particular we prove the following statement:  Let $d\geq 2$, $p\in(d,\infty]$ and $Q$ be a bounded star domain. Then$\displaystyle E_n\Big(W_{p}^{\infty}(Q)\Big)=c(d,p)\Big(\frac {\mathop{mes} Q}{2^d}\Big)^{\frac 1 d +\frac 1 {p'}}\cdot \frac{1+o(1)} {n^{\frac 1 d}}$ $(n\to\infty),$ where $E_n(X):=\inf\Big\{\inf\big\{ e(X,\Phi,x_1,\dots,x_n)\colon\, \Phi\colon\mathbb{R}^n\to\mathbb{R}\big\}\colon x_1,\dots,x_n\in Q\big\},$$e(X, \Phi, x_1,\dots,x_n):= \sup\Big\{\Big|\,\int\limits_{Q}f(x)dx - \Phi(f(x_1),\ldots,f(x_n))\Big|\colon f\in X\Big\}$for $X=W_{p}^{\infty}(Q)$, and $c(d,p)\in \mathbb{R}$ depends only on $d$ and $p$.