On a property of the arithmetic means of monotonous sequences

• Datum: 2022
• Hauptverfasser: Korenovskyi, A. O., Shanin, R. V., Кореновський, А. О., Шанiн, Р. В.
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2022
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/7151
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 For a sequence $Y = \{y_i\}_{i=P+1}^Q$ (the numbers $P, Q \in \mathbb Z$ are fixed, $P < Q$), we consider the arithmetic mean oscillations\begin{equation*}\Omega \big (Y;[p,q] \big )=\frac1{q-p}\sum\limits _{i=p+1}^q\left|y_i-\sigma \big (Y;[p,q] \big )\right|\!,\end{equation*}where $\sigma \big (Y;[p,q] \big )=\displaystyle\frac1{q-p}\sum\nolimits _{i=p+1}^qy_i$ is the arithmetic mean of the sequence $Y$ on the segment $[p,q],$ numbers $P \le p < q \le Q$ are arbitrary.Such oscillations coincide with the integral mean oscillations of the function $f_Y =\sum _{i=P+1}^Qy_i\chi_{(i-1,i)}$ $(\chi_E$ is the characteristic function of the set $E)$$$\Omega(f_Y;[p,q])=\frac1{q-p}\int\limits _p^q\left|f_Y(x)-\sigma(f_Y;[p,q])\right|\,dx,$$ $$\sigma(f_Y;[p,q])=\frac1{q-p}\int\limits _p^qf_Y(x)\,dx,$$on segments with integer boundaries. The main result of the paper is the following equality:\begin{equation*}\max\limits _{ \{p,q\colon P\le p<q\le Q \}}\Omega \big (Y;[p,q] \big ) =\max\limits _{\left\{r\in\mathbb Z\colon P\le r\le Q\right\}}\max\left\{\Omega \big (Y;[P,r] \big ),\Omega \big (Y;[r,Q] \big )\right\},\end{equation*}which holds for every monotonic sequence $Y.$Here, the main point is the fact that the maximum in the right-hand side is taken only over all integer numbers $r.$This equality turns into a well-known equality if we consider the function $f_Y$ instead of the sequence $Y,$ replace the arithmetic mean oscillations by the integral mean oscillations and, in addition, assume that $r$ is not necessarily a integer number.