UDC 517.512.6

MATHEMATICS

V. M. MORDASHEV

APPROXIMATION OF A FUNCTION OF SEVERAL VARIABLES BY A SUM OF FUNCTIONS OF A SMALLER NUMBER OF VARIABLES

(Presented by Academician M. D. Millionshchikov, 3 IV 1968)

Let, in the \(n\)-dimensional parallelepiped
\[ D=\{a_i \leqslant x_i \leqslant b_i,\ i=1,2,\ldots,n\}^* \]
there be given the function
\[ \chi(x_1,x_2,\ldots,x_n)=\chi_1(x_1)\chi_2(x_2)\cdots\chi_n(x_n), \]
where \(\chi_i(x_i)\) does not decrease and
\[ \int_{a_i}^{b_i} d\chi_i(x_i)=1. \]
We consider functions \(\varphi(x_1,x_2,\ldots,x_n)\) that are \(\chi\)-integrable with square in the parallelepiped \(D\).

Denote by \(\sigma_n=\{x_1,x_2,\ldots,x_n\}\) the collection of \(n\) independent variables; \(\sigma_q^i\) (or \(\sigma^i,\ \sigma_q,\ \sigma,\ \tau\)) a subset of \(\sigma_n\); the upper index denotes the number of the subset, the lower one the number of variables constituting this subset; \(\overline{\sigma}\) the set of variables complementing \(\sigma\) to \(\sigma_n\);
\[ \chi(\sigma_q^i)=\chi_{i_1}(x_{i_1})\chi_{i_2}(x_{i_2})\cdots\chi_{i_q}(x_{i_q}),\qquad \sigma_q^i=\{x_{i_1},\ldots,x_{i_q}\}; \]
\[ \varphi(\sigma_n)=\varphi(x_1,x_2,\ldots,x_n),\qquad f(\sigma_q^i)=f(x_{i_1},x_{i_2},\ldots,x_{i_q}) \]
and so on;
\[ \int_{D_{\sigma_q}} f(\sigma_p)\,d\chi(\sigma_q) \]
is the Lebesgue–Stieltjes integral of \(f(\sigma_p)\) over the section
\[ D_{\sigma_q}=\{a_i\leqslant x_i\leqslant b_i,\ x_i\in\sigma_q\} \]
of the parallelepiped \(D\) by the corresponding \(q\)-dimensional hyperplane; in what follows such integration will be called the averaging of the function \(f(\sigma_p)\) over the set of variables \(\sigma_q\).

Lemma 1 and Theorem 2 are proved on the basis of the following obvious properties of the averaging operation:

1.
\[ f(\sigma)=\int_{D_{\overline{\sigma}}} f(\sigma)\,d\chi(\overline{\sigma}). \]

2.
\[ \int_{D_{\sigma}} f(\tau)\,d\chi(\sigma) = \int_{D_{\sigma^2}} \left\{ \int_{D_{\sigma^1}} f(\tau)\,d\chi(\sigma^1) \right\} d\chi(\sigma^2), \qquad \sigma^1\cup\sigma^2=\sigma. \]

3.
\[ \int_{D_{\sigma}}\sum_{i=1}^{k} f_i(\tau^i)\,d\chi(\sigma) = \sum_{i=1}^{k}\int_{D_{\sigma}} f_i(\tau^i)\,d\chi(\sigma), \qquad k<\infty. \]

4.
\[ \int_{D_{\sigma}} f(\tau)\,d\chi(\tau)\geqslant 0, \quad \text{if } f(\tau)\geqslant 0^{**}. \]

Lemma 1. If the function \(\varphi(\sigma_n)\) in the parallelepiped \(D\) is a sum of functions of sets of variables \(\sigma^1,\sigma^2,\ldots,\sigma^m\) that do not contain one another:
\[ \varphi(\sigma_n)=\sum_{i=1}^{m} f_i(\sigma^i), \qquad \sigma^i\not\subseteq\sigma^j \text{ for } i\ne j;\quad 1\leqslant i,j\leqslant m, \]

* \(a_i,\ b_i\) are not necessarily finite.
** The dot above the signs \(\geqslant\) and \(=\) denotes almost everywhere with respect to the measure \(\chi\).

then

\[ \varphi(\sigma_n)= \sum_{1\le i\le m}\varphi(\sigma^i)- \sum_{1\le i<j\le m}\varphi(\sigma^i\cap\sigma^j)+\ldots+ (-1)^{m+1}\varphi(\sigma^1\cap\sigma^2\cap\ldots\cap\sigma^m), \]

where by \(\varphi(\sigma^i)\), \(\varphi(\sigma^i\cap\sigma^j)\), etc., are denoted the averages of \(\varphi(\sigma_n)\) with respect to the corresponding sets of variables \(\sigma^i\), \(\sigma^i\cap\sigma^j\), etc.

Theorem 1. The best mean-square approximation in \(D\), with respect to the measure \(\chi(\sigma_n)\), of the function \(\varphi(\sigma_n)\), for prescribed combinations of variables \(\sigma^1,\sigma^2,\ldots,\sigma^m\) in the approximating sum, is given by the function

\[ \psi(\sigma_n)= \sum_{1\le i\le m}\varphi(\sigma^i)- \sum_{1\le i<j\le m}\varphi(\sigma^i\cap\sigma^j)+\ldots+ (-1)^{m+1}\varphi(\sigma^1\cap\sigma^2\cap\ldots\cap\sigma^m), \]

and, moreover,

\[ \int_D \{\varphi(\sigma_n)-\psi(\sigma_n)\}^2\,d\chi(\sigma_n) = \int_D \varphi^2(\sigma_n)\,d\chi(\sigma_n) - \int_D \psi^2(\sigma_n)\,d\chi(\sigma_n). \]

The results obtained may be used for the analysis and approximate representation of functions of many variables arising in applications.

Recently a number of interesting works of a theoretical-functional nature have been published, for example \((^1\text{--}^3)\), concerning the problem of approximating functions of many variables by sums of functions of a smaller number of variables, in which the existence and uniqueness of the best approximating function are proved.

The author expresses deep gratitude to Acad. M. D. Millionshchikov, V. A. Kholakov, and S. B. Stechkin for useful discussions and criticism.

Institute of Atomic Energy
named after I. V. Kurchatov

Received
21 III 1968

CITED LITERATURE

\(^1\) M.-B. A. Babaev, Izv. AN AzerbSSR, No. 3, 8 (1965).
\(^2\) M.-B. A. Babaev, in: Functional Analysis, Baku, 1965, p. 13.
\(^3\) M.-B. A. Babaev, Dokl. AN AzerbSSR, 23, No. 1, 3 (1967).