Scientific Journals and Yearbooks Published at SAS

We investigate the subsets of the Fr\'echet space $s$ of all sequences of real numbers equipped with the Fr\'echet metric $\rho$ from the Baire category point of view. In particular, we concentrate on the ``convergence" sets of the series $\sum f_n \left(x_n\right)$ that is, sets of sequences $x=(x_n)$ for which the series converges, or has a sum (perhaps infinite), or oscillates. Provided all $f_n$ are continuous real functions, sufficient conditions are given for the ``convergence" sets to be of the first Baire category or residual in $s$.

There are many kinds of the generalization of continuity. T. \v Sal\'at raised the question: Can everywhere discontinuous solution of Cauchy functional equation $$ f(x+y)=f(x)+f(y) $$ be continuous in some generalized sense? The paper deals with this question.

The algebraic or lattice operations in the classes of cliquish or quasicontinuous functions are well known [Z. Grande: \textit{On the maximal multi plicative family for the class of quasicontinuous functions}, Real Anal. Exchange \textbf{15} \mbox{(1989–1990)}, 437–441, Z. Grande, L. Soltysik: \textit{Some remarks on quasicontinuous real functions}, Problemy Mat. \textbf{10} (1990), 79–86]. This also pertains to the symmetrical quasicontinuity or symmetrical cliquishness [Z. Grande: {\it{On the maximal additive and multiplicative families for the quasicontinuities of Piotrowski and Vallin}}, Real Anal. Exchange \textbf{32} (2007), 511–518]. In this article, we examine the superpositions $F(f,g)$, where $F$ is a continuous operation and $f,g$ are cliquish (symmetrically cliquish) or $f$ is continuous ($f$ is symmetrically quasicontinuous with continuous sections) and $g$ is quasicontinuous (symmetrically quasicontinuous).

We present a method of solving functional equations of the type

$$ F(x)-F(y)=(x-y)\bl[b₁f(α₁x+β₁y) +…+b_nf(α_nx+β_ny)\br], $$

where $f,F\colon P\to P$ are unknown functions acting on an integral domain $P$ and parameters $b₁,…,b_n;α₁,…,α_n;β₁,…,β_n\in P$ are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation \begin{equation} 8\bl[F(x)-F(y)\br]=(x-y)\Bgl[f(x)+3f(((x+2y) / (3)))+ 3f(((2x+y) / (3)))+f(y)\Bgr] \label{simp} \end{equation} for $f,F\colon \Rz\to\Rz$ is solved without any regularity assumptions. It is worth noting that (\ref{simp}) stems from a well-known quadrature rule used in numerical analysis.

A Kurzweil-Henstock type integral defined on a zero-dimensional compact abelian group is studied and used to obtain a generalization of some results related to the problem of recovering, by generalized Fourier formulae, the coefficients of convergent series with respect to the characters of such a group.

Generalized inversion formulae for multiplicative integral transform with a kernel defined by characters of a locally compact zero-dimensional abelian group are obtained using a Kurzweil-Henstock type integral.

Special cases of the functional equation \[ h_{1}\left(\frac{x}{c\left(y\right)}\right)\frac{1}{c\left(y\right)}f_{Y}\left(y\right)= h_{2}\left(\frac{y}{d\left(x\right)}\right)\frac{1}{d\left(x\right)}f_{X}\left(x\right) \] are investigated for almost all $\left(x,y\right)\in\r^{2}_{+}$, for the given functions $c$, $d$ and the unknown functions $h_{1}$, $h_{2}$, $f_{X}$ and $f_{Y}$.

This note is closely related to the paper [R. Pinciroli: {\it{On the independence of a generalized statement of Egoroff's theorem from ZFC after T. Weiss,}} Real Anal. Exchange \textbf{32} (2006–2007), 225–232] and it presents slight improvements of its results. Theorem 1.13 shows a connection with Galois-Tukey embeddings; Corollary 1.14 presents another inequality which is dual to the previously known one; Corollary 3.5 shows that there is no distinction between positive outer measure and full outer measure in the given context; and Corollary 4.3 unifies the known counterexamples.

We consider a multimeasure with the Radon-Nikodym derivative and apply its Castaing representation to get a representation of the multimeasure.

Using partitions of the unity ($(PU)$-partition), a new definition of an integral is given for a function $f\colon[a,b] \rightarrow X$, where $X$ is a Banach space, and it is proved that this integral is equivalent to the Bochner integral.

A characterization of the set of discontinuities for different density-type-approximately continuous functions is given.

In this paper, we introduce the class of $[λ, ρ]$-continuous functions. First, we investigate some relations between the classes of $[λ₁, ρ₁]$ and $[λ₂, ρ₂]$-continuous functions for different pairs $[λ₁, ρ₁]$, $[λ₂, ρ₂]$. Next, we give an equivalent condition for a function in order that it will belong to the discussed class.

Let $C_0$ denote the set of all non-decreasing continuous functions $f\colon (0, 1] \to (0, 1]$ such that $\lim_{x\to 0^+}f(x) =0$ and $f(x) \leq x$ for $x\in (0, 1]$ and let~$A$ be a measurable subset of the plane. We define the notion of a density point of $A$ with respect to $f$\!. This is a starting point to introduce the mapping $D_f$ defined on the family of all measurable subsets of the plane, which is so-called lower density. The mapping $D_f$ leads to the topology $\mathcal T_f$, analogously as for the density topology. The properties of the topologies $\mathcal T_f$ are considered.

In the present paper, we introduce the notion of classes of $ρ$-upper continuous functions. We show that $ρ$-upper continuous functions are Lebesgue measurable and, for $ρ<((1) / (2))$, may not belong to Baire class 1. We also prove that a function with Denjoy property can be non-measurable.

We present a Helly type theorem for sequences of functions with values in metric spaces and apply it to representations of some mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. More concretely, a representation of certain majored linear operators on the space of continuous functions into a complete metric space.