On a property of the arithmetic means of monotonous sequences

Abstract

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.