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Approximation of functions and their conjugates in Lp and uniform metric by Euler means

Volosivets S. S., Tyuleneva A. A.

Demonstratio Mathematica

2018

Vol. 51, nr 1

141--150

For 2π-periodic functions from Lp (where 1 < p < ∞) we prove an estimate of approximation by Euler means in Lp metric generalizing a result of L. Rempuska and K. Tomczak. Furthermore, we show that this estimate is sharp in a certain sense. We study the uniform approximation of functions by Euler means in terms of their best approximations in p-variational metric and also prove the sharpness of this estimate under some conditions. Similar problems are treated for conjugate functions.

Weak solutions of fractional order differential equations via Volterra-Stieltjes integral operator

El-Sayed A. M. A., El-Sayed W. G., Abd El-Mowla A. A. H.

Journal of Mathematics and Applications

2017

Vol. 40

85--96

The fractional derivative of the Riemann-Liouville and Caputo types played an important role in the development of the theory of fractional derivatives, integrals and for its applications in pure mathematics ([18], [21]). In this paper, we study the existence of weak solutions for fractional differential equations of Riemann-Liouville and Caputo types. We depend on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type in reflexive Banach spaces.

On regulated functions

Banaś J., Kot M.

Journal of Mathematics and Applications

2017

Vol. 40

21--36

The main purpose of this review article is to present the concept of a regulated function and to indicate the connection of the class of regulated functions with other significant classes of functions. In particular, we give a characterization of regulated functions in terms of step functions and we show that the linear space of regulated functions forms a Banach space under the classical supremum norm.

On bi-dimensional second μ-variation

Ereú J., Giménez J., Merentes N.

Demonstratio Mathematica

2014

Vol. 47, nr 4

910--932

In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in R2. Given a strictly increasing function μ, defined in a closed real interval, we introduce the class BVμ,2 (Iba), of functions of bounded second μ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVμ,2 (Iba) and finally we exhibit a relation between this class and the one of double Riemann–Stieltjes integrals of functions of bi-dimensional bounded variation.