# Scientific Journals and Yearbooks Published at SAS

## Article List

## Tatra Mountains Mathematical Publications

Volume 44, 2009, No. 3

Content:

- Šalát, T. - Vadovič, P.
**The structure of the Fr\'echet space $s$ regarding the series $\sum f_n\left(x_n\right)$.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 1-8. - Kostyrko, P.
**Cauchy functional equation and generalized continuity.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 9-14. - Grande, Z. - Strońska, E.
**Some continuous operations on pairs of cliquish functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 15-25. - Koclęga-Kulpa, B. - Szostok, T. - Wąsowicz, S.
**Some functional equations characterizing polynomials.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 27-40. - Skvortsov, V. - Tulone, F.
**Kurzweil-Henstock type integral in Fourier analysis on compact zero-dimensional group.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 41-51. - Tulone, F.
**Inversion formulae for the integral transform on a locally compact zero-dimensional group.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 53-64. - Lajkó, K. - Mészáros, F.
**Functional equations stemming from probability theory.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 65-80. - Repický, M.
**Generalized Egoroff's theorem.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 81-96. - Olko, J.
**On representation of multimeasure.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 97-103. - Riccobono, G.
**An integral for a Banach valued function.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 105-113. - Horbaczewska, G.
**The set of discontinuities of density-type-approximately continuous functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 115-127. - Nowakowska, K.
**On the family of $[λ,ρ]$-continuous functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 129-138. - Wagner-Bojakowska, E. - Wilczyński, W.
**Density topologies on the plane between ordinary and strong.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 139-151. - Kowalczyk, S. - Nowakowska, K.
**A note on $ρ$-upper continuous functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 153-158. - Duchoň, M. - Maličký, P.
**A Helly theorem for functions with values in metric spaces.**

In*Tatra Mountains Mathematical Publications*. Vol. 44, no. 3 (2009), p. 159-168.

The structure of the Fr\'echet space $s$ regarding the series $\sum f_n\left(x_n\right)$Fulltext Tibor Šalát, Peter Vadovič Fr\'echet space, Baire category, continuous functions, residual set, series.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 $s$.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 1-8. | ||||||

Cauchy functional equation and generalized continuityFulltext Pavel Kostyrko Cauchy functional equation, generalized continuity 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 9-14. | ||||||

Some continuous operations on pairs of cliquish functionsFulltext Zbigniew Grande, Ewa Strońska quasicontinuity, symmetrical quasicontinuity,
cliquishness, symmetrical cliquishness,
strong quasicontinuity, strong
cliquishness, continuous operations,
maximal family for operationThe 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).
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 15-25. | ||||||

Some functional equations characterizing polynomialsFulltext Barbara Koclęga-Kulpa, Tomasz Szostok, Szymon Wąsowicz functional equations on integral domain, quadrature rules, polynomial functionsWe present a method of solving functional equations of the type
$$ F(x)-F(y)=(x-y)\bl[b _{1},…,b_{n};α_{1},…,α_{n};β_{1},…,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.Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 27-40. | ||||||

Kurzweil-Henstock type integral in Fourier analysis on compact zero-dimensional groupFulltext Valentin Skvortsov ^{1)}, Francesco Tulone ^{2)}
compact zero-dimensional abelian group, characters of a group,
Kurzweil-Henstock integral, Perron integral,
Fourier series, coefficient problemA 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 41-51. | ||||||

Inversion formulae for the integral transform on a locally compact zero-dimensional groupFulltext Francesco Tulone ^{1)}
locally compact zero-dimensional abelian group,
characters of a group, Kurzweil\discretionary-Henstock integral,
Fourier series,
multiplicative integral transform, inversion formulaGeneralized 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 53-64. | ||||||

Functional equations stemming from probability theoryFulltext Károly Lajkó, Fruzsina Mészáros characterizations of probability distributions, measurable solution a.e. 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 $f_{Y}$.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 65-80. | ||||||

Generalized Egoroff's theoremFulltext Miroslav Repický Egoroff's theorem, measure, category, cardinal invariants, Galois-Tukey embeddingsThis 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 81-96. | ||||||

On representation of multimeasureFulltext Jolanta Olko ^{1)}
multimeasure, representation, Aumann integralWe consider a multimeasure with the Radon-Nikodym derivative and apply its Castaing representation to get a representation of the multimeasure.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 97-103. | ||||||

An integral for a Banach valued functionFulltext Giuseppa Riccobono ^{1)}
Banach valued function,
$(PU)$-partition, $(PU)^*$-integral, Bochner-integralUsing 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 105-113. | ||||||

The set of discontinuities of density-type-approximately continuous functionsFulltext Grażyna Horbaczewska discontinuities, density topologiesA characterization of the set of discontinuities for different density-type-approximately continuous functions is given.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 115-127. | ||||||

On the family of $[λ,ρ]$-continuous functionsFulltext Katarzyna Nowakowska ^{1)}
density of a set at a point, continuous functions,
approximately continuous functions, path continuityIn this paper, we introduce the class of $[λ, ρ]$-continuous functions. First, we investigate some relations between the classes of $[λ
_{1}, ρ_{1}]$ and $[λ_{2}, ρ_{2}]$-continuous functions for different pairs $[λ_{1}, ρ_{1}]$, $[λ_{2}, ρ_{2}]$. Next, we give an equivalent condition for a function in order that it will belong to the discussed class.Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 129-138. | ||||||

Density topologies on the plane between ordinary and strongFulltext Elżbieta Wagner-Bojakowska, Władysław Wilczyński density point, density topology, density point with respect to $f$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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 139-151. | ||||||

A note on $ρ$-upper continuous functionsFulltext Stanisław Kowalczyk ^{1)}, Katarzyna Nowakowska ^{2)}
density of a set at a point,
continuous functions, measurable functions, path continuity,
Denjoy property, Baire class 1In 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 153-158. | ||||||

A Helly theorem for functions with values in metric spacesFulltext Miloslav Duchoň, Peter Maličký metric space, vector space, vector function,
bounded variation, majored operatorWe 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.
Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 159-168. |