Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces

We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds. We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis.

ON SOME ASPECTS OF VECTOR MEASURES

This paper addresses some properties of vector measures (Banach space valued measures) as well as topological results on some spaces of vector measures of bounded variation.

Differentiable functions and their Fourier coefficients

In the paper we consider the properties of Fourier coefficients of functions that possess derivatives of bounded variation. We investigate the convergence of the special series of Fourier coefficients with respect to general orthonormal systems (ONS). The obtained results are the best possible. We also describe the behavior of subsequences of general ONS.

Stabilization of port-Hamiltonian systems with discontinuous energy densities

<p style='text-indent:20px;'>We establish an exponential stabilization result for linear port-Hamiltonian systems of first order with quite general, not necessarily continuous, energy densities. In fact, we have only to require the energy density of the system to be of bounded variation. In particular, and in contrast to the previously known stabilization results, our result applies to vibrating strings or beams with jumps in their mass density and their modulus of elasticity.</p>

Generalization of Szász operators: quantitative estimate and bounded variation

Difference of exponential type Szász and Szász-Kantorovich operators is obtained. Similar estimates are given for higher order $\mu$-derivatives of the Szász operators and the Szász-Kantorovich type operators acting on the same order $\mu$-derivative of the function. These differences are given in quantitative form using first modulus of continuity. Convergence in variation of the operators in the space of functions with bounded variation with respect to the variation seminorm is obtained. The results propose a general framework covering the results provided by previous literature.

THE EFFECT OF THE WEYL FRACTIONAL INTEGRAL ON FUNCTIONS

This paper mainly discusses the influence of the Weyl fractional integrals on continuous functions and proves that the Weyl fractional integrals can retain good properties of many functions. For example, a bounded variation function is still a bounded variation function after the Weyl fractional integral. Continuous functions that satisfy the Holder condition after the Weyl fractional integral still satisfy the Holder condition, furthermore, there is a linear relationship between the order of the Holder conditions of the two functions. At the end of this paper, the classical Weierstrass function is used as an example to prove the above conclusion.

Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

A NEW KIND OF THE VARIANT OF THE MODIFIED BERNSTEIN-KANTOROVICH OPERATORS DEFINED BY ¨OZARSLAN AND DUMAN

In the present article, we dene a new kind of the modified Bernstein-Kantorovich operators defined by ¨ Ozarslan (https://doi.org/10.1080/01630563.2015.1079219) i.e. we introduce a new function ς(x) in the modified Bernstein-Kantorovich operators defined by Ozarslan with the property ({) is an infinitely differentiable function on [0; 1]; ς(0) = 0; ς(1) = 1 and ς’(x) > 0 for all x∈ [0; 1]. We substantiate an approximation theorem by using of the Bohman-Korovkins type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions for the our operators and the rate of convergence of functions by means of derivatives of bounded variation are also studied. We study an approximation theorem with the help of Bohman-Korovkins type theorem in A-Statistical convergence. Lastly, by means of a numerical example, we illustrate the convergence of these operators to certain functions through graphs with the help of MATHEMATICA and show that a careful choice of the function ς(x) leads to a better approximation results as compared to the modified Bernstein-Kantorovich operators defined by Ozarslan (https://doi.org/10.1080/01630563.2015.1079219).

The form of locally defined operators in waterman spaces

A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.