Topic
Regulated function
About: Regulated function is a research topic. Over the lifetime, 38 publications have been published within this topic receiving 291 citations.
Papers
1 Jan 1976
TL;DR: In this paper, it was shown that there is a nontrivial class of regulated functions, each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function.
Abstract: It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded -variation for convex $. The regulated functions on a closed interval are those functions whose right and left limits exist at each point. Every regulated function is bounded, has a countable set of discontinuities, and is the limit of a uniformly convergent sequence of step functions. The regulated functions are of importance in the theory of stochastic processes and in the theory of everywhere convergence of Fourier series. Functions of bounded variation are regulated, as are the functions of bounded 4>-variation (1) and the functions of bounded A-variation (3). In §1 we shall show that each regulated function is of bounded -variation for some 4>. A function of bounded variation has a canonical representation as the sum of a continuous function of bounded variation and a sum of jump functions, the jumps being those of the given function of bounded variation. In §2 we shall investigate the possibility that a regulated function have a representation as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. We shall see that, although we can characterize a nontrivial class of functions for which this representation is possible, we can construct functions which have no such representation. 1. Let (0) = 0, $(x) > 0 for x > 0. A function / defined on an interval / is said to be of bounded $- variation ( - BV) if the «^-variation of/,
32 citations
TL;DR: In this article, for a real cadlag function f and a positive constant c, the smallest total variation possible among all functions uniformly approximating f with accuracy c/2 was found.
Abstract: For a real cadlag function f and a positive constant c we find another cadlag function, which has the smallest total variation possible among all functions uniformly approximating f with accuracy c/2. The solution is expressed with the truncated variation, upward truncated variation and downward truncated variation introduced in [L1] and [L2]. They are are always finite even if the total variation of f is infinite, and they may be viewed as the generalisation of the Hahn-Jordan decomposition for real cadlag functions. We also present partial results for more general functions.
18 citations
1 Jan 1991
TL;DR: In this paper, boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with, and the adjoint problems are introduced in such a way that the usual duality theorems are valid.
Abstract: Boundary value problems for generalized linear differential equations and the corresponding controllability problems are dealt with. The adjoint problems are introduced in such a way that the usual duality theorems are valid. As a special case the interface boundary value problems are included. In contrast to the earlier papers by the author the right-hand side of the generalized differential equations as well as the solutions of this equation can be in general regulated functions (not necessarily of bounded variation). Similar problems in the space of regulated functions were treated e.g. by Ch. S. Honig, L. Fichmann and L. Barbanti, who made use of the interior (Dushnik) integral. In this paper the integral is the Perron-Stieltjes (Kurzweil) integral.
17 citations
TL;DR: In this article, the concept of Δ-sub-derivative on time scales was introduced to define e-equivalent impulsive functional dynamic equations on almost periodic time scales.
Abstract: In this paper, we introduce the concept of Δ-sub-derivative on time scales to define e-equivalent impulsive functional dynamic equations on almost periodic time scales. To obtain the existence of solutions for this type of dynamic equation, we establish some new theorems to characterize the compact sets in regulated function space on noncompact intervals of time scales. Also, by introducing and studying a square bracket function $[x(\cdot),y(\cdot) ]:\mathbb{T}\rightarrow\mathbb{R}$
on time scales, we establish some new sufficient conditions for the existence of almost periodic solutions for e-equivalent impulsive functional dynamic equations on almost periodic time scales. The final section presents our conclusion and further discussion of this topic.
16 citations
TL;DR: In this paper, the play operator is linked with truncated variation functionals, and a semi-explicit expression for play operator in terms of these functionals is provided, which gives the best possible lower bounds for the total variation of the outputs of the play operators and its Jordan-like decomposition.
Abstract: The play operator minimalizes the total variation on intervals $[0,T], T> 0$, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article we link the play operator with so called truncated variation functionals, introduced recently by the second-named author, and provide a semi-explicit expression for the play operator in terms of these functionals. Generalisation for time-dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan-like decomposition.
16 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 1 |