Accuracy of the approximation of the characteristic functions by polynomials
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 141-144
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In the paper one obtains a series of statements allowing us to estimate the accuracy of the approximation of the characteristic function $f(t)=\int e^{itx}dV(x)$ by a polynomial of integer powers of $it$. For example,
$$
C_1\Gamma(b)\le\sup_{|t|\le b}|f(t)-1-\sum_{l=1}^{2M-1}\frac{(it)^l}{l!}d_l|\le C_2\Gamma(b)
$$
where the positive constants $$, $$ depend only on $M$, $M\ge1$ is an integer, $b>0,$
$$
\Gamma(b)=\int_{-\infty}^{\infty}\min\Big(1, (xb)^{2M}\Big)dV(x)+\max_{1\le l\le2M}b^2|d_l-\int_{|xb|\le1}x^ldV(x)|.
$$
@article{ZNSL_1985_142_a14,
author = {L. V. Rozovskii},
title = {Accuracy of the approximation of the characteristic functions by polynomials},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {141--144},
publisher = {mathdoc},
volume = {142},
year = {1985},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a14/}
}
L. V. Rozovskii. Accuracy of the approximation of the characteristic functions by polynomials. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 141-144. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a14/