On One Property of the Mean-Arithmetic Oscillations of a Monotone Sequence

For a number sequence $Y={\left\{{y}_i\right\}}_{i=P+1}^Q$ ( the numbers P,Q ∈ ℤ are fixed and P < Q), we consider the mean-arithmetic oscillations

$$ \Omega \left(Y;\left[p,q\right]\right)=\frac{1}{q-p}\sum \limits_{i=p+1}^q\left|{y}_i-\sigma \left(Y;\left[p,q\right]\right)\right|, $$

where $\sigma \left(Y;\left[p,q\right]\right)=\frac{1}{q-p}{\sum}_{i=p+1}^q{y}_i$ is the arithmetic mean of the sequence Y on the segment [p, q] and P ≤ p < q ≤ Q are arbitrary numbers. These oscillations coincide with the mean-integral oscillations of the function ${f}_Y={\sum}_{i=P+1}^Q{y}_i{\chi}_{\left(i-1,i\right)}$ (𝜒_E is the characteristic function of the set E) :

$$ \Omega \left({f}_Y;\left[p,q\right]\right)=\frac{1}{q-p}\int_p^q\left|{f}_Y(x)-\sigma \left({f}_Y;\left[p,q\right]\right)\right| dx,\kern1em \sigma \left({f}_Y;\left[p,q\right]\right)=\frac{1}{q-p}\int_p^q{f}_Y(x) dx, $$

on segments with integer boundaries.

The main result of the paper is the following equality:

$$ \underset{\left\{p,q{\kern0.5em} :{\kern1em} P\le p<q\le Q\right\}}{\max}\Omega \left(Y;\left[p,q\right]\right)=\underset{\left\{r\in \mathbb{Z}{\kern0.5em} :{\kern1em} P\le r\le Q\right\}}{\max}\max \left\{\Omega \left(Y;\left[P,r\right]\right),\Omega \left(Y;\left[r,Q\right]\right)\right\}, $$

which is true for any monotone sequence Y. In this case, the main point is the fact that the maximum on the right-hand side is taken only over all integer numbers r. This equality turns into a well-known equality if we take a function f_Y instead of the sequence Y, replace the mean-arithmetic oscillations by the mean-integral oscillations and, in addition, assume that the number r on the right-hand side is not necessarily integer.