Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 759-767
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O. P. Filatov. Evaluation of the limits of maximal means. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 759-767. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a11/
@article{MZM_1996_59_5_a11,
author = {O. P. Filatov},
title = {Evaluation of the limits of maximal means},
journal = {Matemati\v{c}eskie zametki},
pages = {759--767},
year = {1996},
volume = {59},
number = {5},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a11/}
}
TY - JOUR
AU - O. P. Filatov
TI - Evaluation of the limits of maximal means
JO - Matematičeskie zametki
PY - 1996
SP - 759
EP - 767
VL - 59
IS - 5
UR - http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a11/
LA - ru
ID - MZM_1996_59_5_a11
ER -
%0 Journal Article
%A O. P. Filatov
%T Evaluation of the limits of maximal means
%J Matematičeskie zametki
%D 1996
%P 759-767
%V 59
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a11/
%G ru
%F MZM_1996_59_5_a11
It is proved that the limit $$ \lim_{\Delta\to\infty}\sup_\gamma\frac 1\Delta \int_0^\Delta f\bigl(\gamma(t)\bigr)\,dt, $$ where $f\colon\mathbb R\to\mathbb R$ is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation $\dot\gamma\in[\omega_1,\omega_2]$, coincides with the limit $$ \lim_{T\to\infty}\sup_{c\ge0}\varphi_f(k,T,c), $$ where $$ \varphi_f=\frac{(k-1)\overline I_f(T,c)} {1+(k-1)\overline\lambda_f(T,c)},\qquad k=\frac{\omega_2}{\omega_1}. $$ Here $\overline\lambda_f=\lambda_f/T$, $\overline I_f=I_f/T$, and $\lambda_f$ is the Lebesgue measure of the set $$ \bigl\{\gamma\in[\gamma_0,\gamma_0+T]: f(\gamma)\ge c\bigr\}=A_f,\qquad I_f=\int_{A_f}f(\gamma)\,d\gamma. $$ It is established that this limit always exists for almost-periodic functions $f$.
[1] Filatov O. P., Khapaev M. M., “O vzaimnoi $\epsilon$-approksimatsii reshenii sistemy differentsialnykh vklyuchenii i usrednennogo vklyucheniya”, Matem. zametki, 47:5 (1990), 127–134 | MR | Zbl
[2] Filatov O. P., Khapaev M. M., “Usrednenie differentsialnykh vklyuchenii s “bystrymi” i “medlennymi” peremennymi”, Matem. zametki, 47:6 (1990), 102–109 | MR | Zbl
