Volume 18, Issue 2

This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p (·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

In 2005, Tkebuchava constructed a set of logarithmic summation methods which includes Riesz and Nörlund logarithmic summation methods as limit cases. He gave estimates for the kernels of a trigonometric system and also some consequences with respect to the convergence of the means of a Fourier series. Here we give such good estimates for the kernels with respect to Walsh and Walsh–Kaczmarz systems as Tkebuchava did.

For a given topological space E , some properties of absolutely nonmeasurable real-valued functions, with respect to a certain class of measures on E , are considered. These functions are compared with Sierpiński–Zygmund type real-valued functions on E which have the property that they are discontinuous on every uncountable subset of E .

We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set on which the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is not the same as the boundedness in weighted grand Lebesgue space.

A generalization of the Euler beta function to the case of a multi-dimensional variable is defined. In this context, the original beta function is a function of a two-dimensional variable. An analogue of the Euler formula for this new function is derived for the case of a three-dimensional variable. Based on the derived formula, a number of relations for the Gauss hypergeometrical function are obtained. Moreover, the analytic formulae for some new integrals of special functions are obtained.

Let ℱ be a family of meromorphic functions in a domain D ; let k ≥ 2 be a positive integer; and let a, b and c be complex numbers such that b ≠ 0 and a ≠ c . If, for each ƒ ∈ ℱ, all zeros of ƒ have multiplicity at least k , ƒ( z ) = a ⇔ D (ƒ) = b , and D (ƒ) = 0 ⇒ ƒ( z ) = c , where D (ƒ) is the differential polynomial of ƒ( z ), then ℱ is normal in D .

The purpose of this paper is to present some fixed point results for self-generalized (singlevalued and multivalued) contractions in ordered metric spaces and in metric spaces endowed with a graph. Our theorems generalize and extend some recent results in the literature.

We consider the cutting process of the generalized Möbius–Listing surface along some closed line which is parallel to its border line. We show connections of the resulting mathematical objects with the set of knots and links. In some cases we count the indices of the corresponding objects according to the known classification.

Variation formulas of solution are proved for a non-linear differential equation with constant delay. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

An admissibility for topological degree of variable Lebesgue, Besov and Triebel–Lizorkin spaces is given.

Let μ be a non-negative Radon measure on which satisfies only the polynomial growth condition. Let 𝒴 be a Banach space and H 1 ( μ ) be the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H 1 ( μ ) to 𝒴 if and only if T maps all ( p, γ )-atomic blocks into uniformly bounded elements of 𝒴; moreover, the authors prove that for a sublinear operator T bounded from L 1 ( μ ) to L 1, ∞ ( μ ), if T maps all ( p, γ )-atomic blocks with p ∈ (1, ∞) and γ ∈ ℕ into uniformly bounded elements of L 1 ( μ ), then T extends to a bounded sublinear operator from H 1 ( μ ) to L 1 ( μ ). For the localized atomic Hardy space h 1 ( μ ), the corresponding results are also presented. Finally, these results are applied to Calderón–Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón–Zygmund operators or fractional integral operators with Lipschitz functions to simplify the existing proofs in the related papers.