Valentin Skvortsov

The essential variation of a function and some convergence theorems

AbstractВВОДИтсь ОпРЕДЕлЕНИ Е ВАРИАцИИ ФУНкцИИ, пР И кОтОРОМ ФОРМУлА

On dyadic integrals and some other integrals associated with local systems

V(F,E) = \int_E {|\bar DF(x)} |dx

On the Coefficients of Multiple Series with Respect to Vilenkin System

спРАВЕДлИВА Дль пРОИ жВОльНОИ ФУНкцИИF И пРОИжВОльНОгО ИжМЕР ИМОгО МНОжЕстВАE НА ОтРЕжкЕ пРьМОИ. В т ЕРМИНАх ЁтОИ ВАРИАцИ И пОлУЧЕНы тЕОРЕМы О пОЧлЕННОМ ДИФФЕРЕНцИРОВАНИИ п ОслЕДОВАтЕльНОстИ Ф УНкцИИ И тЕОРЕМы О пРЕДЕльНОМ пЕРЕхОДЕ пОД жНАкОМ И НтЕгРАлА ДАНжУА-пЕРР ОНА.

On M0-sets for series with respect to characters of compact zero-dimensional

Abstract An Henstock–Kurzweil type integral with respect to a P -adic basis is considered. It is shown that a P -adic basis possesses the Ward property if and only if the sequence by which it is defined is bounded. As a consequence, some full descriptive characterizations of the P -adic integral in the bounded case are obtained. Moreover, an example of an exact P -adic primitive which is not a VBG function and does not satisfy the Lusin condition (N) is constructed.

A Hake-type theorem for integrals with respect to abstract derivation bases in the Riesz space setting

We consider some variational properties of indefinite Henstock-type integrals defined by local systems of sets. We study in particular the dyadic path integral which is a special example of such integrals. We obtain full descriptive characterization of the dyadic path integral in terms of the correspondent variational measure and compare it with the Henstock integral associated with the dyadic interval basis.

THE STRONG PERRON INTEGRAL IN ℝ n REVISITED

Abstract A Kurzweil-Henstock type integral defined on a zero-dimensional compact abelian group is studied and used to obtain a generalization of some results related to the problem of recovering, by generalized Fourier formulae, the coefficients of convergent series with respect to the characters of such a group.

On variational measures related to some bases

Abstract We give a sufficient condition for coefficients of double series Σ Σn,m an,m χn,m with respect to Vilenkin system to be convergent to zero when n + m → ∞. This result can be applied to the problem of recovering coefficients of a Vilenkin series from its sum.

On classes of functions generating absolutely continuous variational measures

Abstract Generalizing our previous result on Walsh system, we consider the system of characters of zero-dimensional compact abelian group. For this system, we give a construction of a perfect M0-set set whose Hausdorff h-measure equals zero, where h is any non-decreasing right-continuous function taking the value zero at the origin.