bel and Euler Summation Formulas for SBV(R)

Internat. J. Math. Math. Sci., 1986

The aim of this article is to unify a large part of present knowledge about the behaviour of Fourier series of functions of generalized bounded variation. The connections between various concepts are discussed, particular attention being payed to those due to Waterman and Chanturiya. Exploiting the existing interactions and utilizing the power of one or another approach to some typical questions of the Fourier theory, a number of previously unnoticed results are obtained in the course of this

Illinois journal of mathematics

This paper is concerned with the relation between the regularity properties of a function and those of its variation function andwith some applications of these relations to some problems about harmonic functions. Before we state our results we recall some definitions and known facts.

Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov's introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if : [ , ] → is of bounded variation then is the sum of an absolute continuous function and a singular function where the total variation of generates a singular measure and is absolute continuous with respect to. In this paper we prove that a function of bounded variation has two representations: one is which was described with an absolute continuous part with respect to the Lebesgue measure , while in the other an integral with respect to forms the absolute continuous part and () defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain [ , ] ].

Canadian Mathematical Bulletin, 1981

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.

2012

We prove an extension of the Ocone–Karatzas integral representation, valid for all BVBV functions on the classical Wiener space. We also establish an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of BVBV composite random variables.

1996

We show that lower semicontinuous functionals defined on De Giorgi and Ambrosio's space of special functions of bounded variation admit an integral representation with Carathéodory integrands, under some growth and continuity conditions.

Nonlinear Analysis-theory Methods & Applications, 2018

Moduli of path families are widely used to study Sobolev functions. Similarly, the recently introduced approximation (AM -) modulus is helpful in the theory of functions of bounded variation (BV ) in R n [14]. We continue this direction of research. Let Γ E be the family of all paths which meet E ⊂ R n . We introduce the outer measure E → AM (Γ E ) and compare it with other (n-1)-dimensional measures. In particular, we show that AM (Γ E ) = 2 H n-1 (E) whenever E lies on a countably (n -1)rectifiable set. Further, we study functions which have bounded variation on AM -a.e. path and we relate these functions to the classical BV functions which have only bounded essential variation on AM -a.e. path. We also characterize sets E of finite perimeter in terms of the AM -modulus of two path families naturally associated with E. VHE and JM have been supported by the grant GA ČR P201/18-07996S of the Czech Science Foundation.

Proyecciones, 2020

Stochastic Processes and their Applications, 2012

We prove an extension of the Ocone-Karatzas integral representation, valid for all BV functions on the classical Wiener space. We establish also an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of BV composite random variables.

Analysis Mathematica, 1996

There are a number of ways of constructing a variation of a real function on a set E C R. The notions of relative and absolute variations, weak and strong variations as defined in are well adapted to functions of bounded variation on a set. Another construction known as "Munroe's method II" (see ), also nicely reflecting variational properties of a function of bounded variation on a set, reveals serious disadvantages if applied for an arbitrary continuous function Our intention in this paper is to consider a definition of variation which, when applied to any function, gives full information on the absolute integrability of its derivates on a measurable set. In fact, this definition makes it possible to generalize, for the case of an arbitrary function and a measurable set, the known equality between the total variation of an absolutely continuous function and the integral of the modulus of the derivative of this function. In particular, this means that in the case of a function of bounded variation F we want our definition to ignore singularities of F. This definition of variation is then applied to establish some convergence theorems for generalized integrals. 1. We introduce some notations. If E C R then IEI denotes the outer Lebesgue measure of E. "Almost everywhere" (abbreviated as a.e.) will be always used in the sense of the Lebesgue measure. An interval is always a compact subinterval of R. A collection of intervals is called nonoverlapping whenever their interiors are disjoint. In this paper A --[a, b] is a fixed interval of R.