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Tatra Mountains Mathematical Publications


Volume 44, 2009, No. 3

Content:


  The structure of the Fr\'echet space $s$ regarding the series $\sum f_n\left(x_n\right)$
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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 continuity
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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 functions
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Zbigniew Grande, Ewa Strońska

quasicontinuity, symmetrical quasicontinuity, cliquishness, symmetrical cliquishness, strong quasicontinuity, strong cliquishness, continuous operations, maximal family for operation

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).

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 15-25.

 
  Some functional equations characterizing polynomials
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Barbara Koclęga-Kulpa, Tomasz Szostok, Szymon Wąsowicz

functional equations on integral domain, quadrature rules, polynomial functions

We present a method of solving functional equations of the type

$$ F(x)-F(y)=(x-y)\bl[b1f(α1x+β1y) +…+bnf(αnx+βny)\br], $$

where $f,F\colon P\to P$ are unknown functions acting on an integral domain $P$ and parameters $b1,…,bn1,…,αn1,…,βn\in P$ 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 group
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Valentin Skvortsov 1), Francesco Tulone 2)

1)Department of Mathematics, Moscow State University, 119899 Moscow, Russia. vaskvor2000@yahoo.com
2)Department of Mathematics, University of Palermo; via Archirafi 34; I–90123 Palermo; ITALY. tulone @math.unipa.it


compact zero-dimensional abelian group, characters of a group, Kurzweil-Henstock integral, Perron integral, Fourier series, coefficient problem

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 41-51.

 
  Inversion formulae for the integral transform on a locally compact zero-dimensional group
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Francesco Tulone 1)

1)Department of Mathematics, University of Palermo; via Archirafi 34; I–90123 Palermo; ITALY. tulone @math.unipa.it


locally compact zero-dimensional abelian group, characters of a group, Kurzweil\discretionary-Henstock integral, Fourier series, multiplicative integral transform, inversion formula

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 53-64.

 
  Functional equations stemming from probability theory
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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 theorem
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Miroslav Repický

Egoroff's theorem, measure, category, cardinal invariants, Galois-Tukey embeddings

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 81-96.

 
  On representation of multimeasure
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Jolanta Olko 1)

1)Institute of Mathematics, Pedagogical University; Podchorążych 2; PL--30-084 Kraków; POLANDi. jolko@ap.krakow.pl


multimeasure, representation, Aumann integral

We 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 function
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Giuseppa Riccobono 1)

1)Dipartimento di Matematica, Universitá di Palermo; Via Archirafi 34; I--90123 Palermo; ITALY. ricco@math.unipa.it


Banach valued function, $(PU)$-partition, $(PU)^*$-integral, Bochner-integral

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 105-113.

 
  The set of discontinuities of density-type-approximately continuous functions
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Grażyna Horbaczewska

discontinuities, density topologies

A 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 functions
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Katarzyna Nowakowska 1)

1)Institute of Mathematics, Academia Pomeraniensis; ul. Arciszewskiego 22b; PL–76-200 Słupsk POLAND. nowakowska k@go2.pl


density of a set at a point, continuous functions, approximately continuous functions, path continuity

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 129-138.

 
  Density topologies on the plane between ordinary and strong
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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 functions
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Stanis­ław Kowalczyk 1), Katarzyna Nowakowska 2)

1)Department of Mathematics, Pedagogical University: Arciszewskiego 22b; PL-76-200 S. stkowalcz@onet.eu
2)Institute of Mathematics, Academia Pomeraniensis; ul. Arciszewskiego 22b; PL–76-200 Slupsk, POLAND. nowakowska k@go2.pl


density of a set at a point, continuous functions, measurable functions, path continuity, Denjoy property, Baire class 1

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 153-158.

 
  A Helly theorem for functions with values in metric spaces
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Miloslav Duchoň, Peter Maličký

metric space, vector space, vector function, bounded variation, majored operator

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.

Tatra Mountains Mathematical Publications. Volume 44, 2009, No. 3: 159-168.

 
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