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The problem of stability for J. Marcinkiewicz's theorem
N. A. Sapogov
Abstract:
In this paper we investigate stability of the well known theorem due to J. Marciniciewicz asserting that
$\exp P(t)$ where
$P(t)$ is a polynomial can be a characteristic function only when the degree of
$P(t)$ is
$\leq2$. Our main result is given by the following theorem.
Theorem. {\it Let
$|\exp P_{2n}(t)-\varphi(t)|\leq\varepsilon,\quad t\in[-T,T]$, where
$$
P_{2n}(t)=-\frac12t^2+\sum_{k=2}^n a_{2k}t^{2k},
\quad a_{2k}\in R^1,\quad|a_{2k}|\leq H,\quad k=2,3,\dots,n,\quad a_{2n}<0
$$
$\varphi(t)=\varphi(-t)$ – even characteristic function. Then
$$
-a_{2n}\leq\frac{k_1\cdot H^{1-1/n}}{(\log1/\varepsilon_2)^{1-1/n}}+
\frac{k_2\cdot H^{1+1/n}}{(\log1/\varepsilon_2)^{1/n}},
$$
if $\varepsilon_2=k[\varepsilon(\log T+1)+T^{-1}(\log T)^{1/2n}]$ is sufficient small;
$K$ is an absolute constant,
$K_1$ and
$K_2$ depend on
$n$ only.}
UDC:
519.2