The stability of solutions of the functional equations connected with characterization theorems for probability distributions

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.