This article is cited in
4 papers
The stability of solutions of the functional equations connected with characterization theorems for probability distributions
N. A. Sapogov
Abstract:
The present paper contains some results on the solutions
$\Psi_j(x)$ of the functional inequality
\begin{equation}
\biggl|\sum\Psi_j(a^T_jt)\biggr|\leq\varepsilon,
\tag{1}
\end{equation}
where $a^T_j=(a_{1j},a_{2j},\dots,a_{pj})\in\mathbb{R}^p$ all the coefficients
$a_{ij}$ are constants,
$t=(t_1,t_2,\dots,t_p)\in\mathbb{R}^p$,
$a_j^Tt=\sum_{i=1}^p a_{ij}t_i$,
$p\geq2$, the relation (1) holds for all
$t_j\in\mathbb{R}^1$,
$j=1,2,\dots,n$. Inequality (1) is connected with certain characterization theorem in theory of probability and statistics. For the sake of simplicity we suppose that
$\Psi_j(x)$ are continuous functions,
$x\in\mathbb{R}^1$. We obtain the following main results
Theorem.
{\it Let (1) holds,
$n\ge 1$,
$p=2$,
$\Delta_{kj}=a_{1j}a_{2k}-a_{1k}a_{2j}\neq0$ for
$j\ne k$,
$\varepsilon>0$
is
an arbitrary positive number. Then there exist polynomials
$P_{n,j}$,
$j=1,\dots,n$, such that
$$
\biggl|\Psi_j(x)-P_{n,j}(x)\biggr|\le 4^{n-2}\varepsilon
$$
for all
$x\in\mathbb{R}^1$,
$j=1,2,\dots,n$. The degrees of
$P_{n,j}(x)$
are
$\leq n-2$.}
The particular case
$n=3$,
$p=2$ is of some interest and was investigated in more details.
UDC:
519.2