Аналіз пульсових хвиль власних векторів оператора диференціювання в базисі перетворення Уолша-Адамара

The opportunity and prospect of the analysis of signals of a pulse wave is shown in the field of orthogonal transformations, for which transformation are of an own vector of the discrete operator of differentiation.Показана возможность анализа сигналов пульсовой волны в области ортогональных преобразований, для которых трансформантами являются собственные вектора дискретного оператора дифференцирования.Показана можливість і перспективність аналізу сигналів пульсової хвилі в області ортогональних перетворень, для яких трансформантами є власні вектори дискретного оператора диференціювання, а оригіналами - трансформанти Уолша - Адамара

Geometric maximal operators and BMO on product bases

We consider the problem of the boundedness of maximal operators on BMO on shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO

On differentiation of integrals with respect to bases of convex sets.

Differentiation of integrals of functions from the class $Lip(1,1)(I^2)$ with respect to the basis of convex sets is established. An estimate of the rate of differentiation is given. It is also shown that there exist functions in $Lip(1,1)(I^N)$, N ≥ 3, and $H^{ω}_{1}(I^2)$ with ω(δ)/δ → ∞ as δ → +0 whose integrals are not differentiated with respect to the bases of convex sets in the corresponding dimension