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

  • O. V. Kovalenko Oles Honchar Dnipro National University
Keywords: cubature formulae, multidimensional Sobolev space, star domain, asymptotically optimal recovery method


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 $p$.

Author Biography

O. V. Kovalenko, Oles Honchar Dnipro National University

Oles Honchar Dnipro National University


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How to Cite
Kovalenko, O. V. (2024). On optimization of cubature formulae for Sobolev classes of functions defined on star domains. Matematychni Studii, 61(1), 84-96. https://doi.org/10.30970/ms.61.1.84-96