Search

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed. [ABSTRACT FROM AUTHOR]

A Banach space X of analytic function in , the unit disc in , is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of. If X and Y are two admissible Banach spaces of analytic functions in and g is a holomorphic function in , g is said to be a multiplier from X to Y if g ·f ∈ Y for every f ∈ X. The space of all multipliers from X to Y is denoted M (X, Y), and M (X) will stand for M (X, X). The closed graph theorem shows that if g ∈ M (X, Y) then the multiplication operator Mg, defined by Mg (f) = g · f, is a bounded operator from X into Y. It is known that M (X) ⊂ H∞ and that if g ∈ M (X), then ∥g∥H∞ ≤ ||Mg||. Clearly, this implies that M (X, Y) ⊂ H∞ if Y ⊂ X. If Y ⊄ X, the inclusion M (X, Y) ⊂ H∞ may not be true. In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M (X, Y) ⊂ H∞ holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c0(κ) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c0(κ) and c0(I) are either subspaces of ℓ∞(κ) or contain a complemented copy of c0(κ+); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c0 and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary, which solves a problem of Castillo and Papini. [ABSTRACT FROM AUTHOR]

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in p spaces is proven. Further, we derive the fractal analogues of some p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed. [ABSTRACT FROM AUTHOR]

Let C(X) denote the ring of all real-valued continuous functions on a topological space X; and C∞(X) be the subring of all functions C(X) which vanish at infinity. In [2], the paper "Rings of continuous functions vanishing at infinity," (Comment. Math. Univ. Carolin.45(3) (2004), 519–533), by A.R. Aliabad, F. Azarpanah, and M. Namdari, it is shown that for every completely regular Hausdorff space X, whenever C∞(X) ≠ (0), then there exists a locally compact space Y such that C∞(X) ≅ C∞(Y). In fact, the space Y may be considered as a nonempty open locally compact subspace of X. In the present paper, analogous results are derived in a pointfree context in which topological spaces are replaced by frames. [ABSTRACT FROM AUTHOR]

In the paper by M. Filipczak, G. Ivanova and J. Wódka, Comparison of some families of real functions in porosity terms, Math. Slovaca67(5) (2017), 1155–1164, it was proved that the family of quasi-continuous Świątkowski functions is porous but not strongly porous in the space of quasi-continuous functions. We improved this result. We found the unique number q = 2/3 such that the family of quasi-continuous Świątkowski functions is lower q-porous in this space. [ABSTRACT FROM AUTHOR]

In the paper, for certain classes of operators F in the space of analytic functions, we prove the discrete universality for compositions F (ζ(s, α; , )), where ζ(s, α; ; ) is a collection consisting from periodic and periodic Hurwitz zeta-functions, i. e., the approximation of analytic functions by discrete shifts F (ζ(s + ikh, α; ; )) with h > 0 and k = 0, 1,.... For this, a theorem of [12] is applied. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the T3½ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space T via Wallman compactifiaction Wall(T), we constuct a measure of non-compactness for the space Cb(T). We also study some of the properties of this object and give another version of Darbotype theorem, suitable for this particular case. [ABSTRACT FROM AUTHOR]

In the paper, we recall the Wallman compactification of a Tychonoff space T (denoted by Wall(T)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between Cb(T), the space of continuous and bounded functions on T, and C(Wall(T)), the space of continuous functions on the Wallman compactification of T. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-Čech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultrafilters. [ABSTRACT FROM AUTHOR]

The paper is an overview of selected results on weaker forms of classical selection principles of Menger, Hurewicz, Rothberger and Gerlits-Nagy obtained in the last few years. We review some very recent results on star selection principles, an important subfield of selection principles theory (including boundedness properties in function spaces considered as uniform spaces and topological groups), ideal versions of Hurewicz-type properties, selection principles called weak, almost, quasi, and combinatorics of covers by generalized open sets. A number of examples which clarify relationships among properties we consider is given, and some open questions are posed. [ABSTRACT FROM AUTHOR]

The finite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over (–1, 1) having non-trivial Boyd indices. For X = Lp, 1 < p < ∞, this operator has been intensively investigated since the 1940's (also under the guise of the "airfoil equation"). Recently, the extension and inversion of T : X → X for more general X has been studied in [6], where it is shown that there exists a larger space [T, X], optimal in a well defined sense, which contains X continuously and such that T can be extended to a continuous linear operator T : [T, X] → X. The purpose of this paper is to continue this investigation of T via a consideration of the X-valued vector measure mX : A.→ induced by T and its associated integration operator. In particular, we present integral representations of T: X →X based on the L1-space of mX and other related spaces of integrable functions. [ABSTRACT FROM AUTHOR]

In this paper we study nonlinear diameter preserving mappings defined between function spaces and obtain generalizations of, basically, all known results concerning diameter preservers. In particular, we give a complete description for algebras of continuously differentiable functions, (little) Lipschitz algebras and dense function spaces. [ABSTRACT FROM AUTHOR]

In this paper we continue previous investigations on the weaker forms of the Menger property in bitopological spaces. We introduce weakly Menger property and study some topological properties of almost and weakly Menger bitopological spaces. We also consider the almost Hurewicz spaces in a bitopological context. [ABSTRACT FROM AUTHOR]

In this paper we investigate the linear and affine mappings of near-vector spaces. [ABSTRACT FROM AUTHOR]

In this paper, for a frameL, we characterize modified sobriety in stratifiedL-topological spaces and strongL-topological spaces internally using certain classes of stratifiedL-filters. While the first characterization using completely primeL-filters is trivial and applies to all stratifiedL-topological spaces, the other one using irreducibleL-filters generalizes an approach of R.E. Hoffmann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strongL-topological spaces. [ABSTRACT FROM AUTHOR]