Real Analysis Exchange Articles (Project Euclid)

How to Concentrate Idempotents

Thu, 05 Aug 2010 15:41 EDT

J. Marshall Ash

Source: Real Anal. Exchange, Volume 35, Number 1, 1--20.

Abstract:
Call a sum of exponentials of the form $f(x)=\exp\left( 2\pi iN_{1}x\right) +\exp\left( 2\pi iN_{2}x\right) +\cdot\cdot\cdot+\exp\left( 2\pi iN_{m}x\right) $, where the $N_{k}$ are distinct integers, an \textit{idempotent}. We have $L^{p}$\textit{ interval concentration} if there is a positive constant $a$, depending only on $p$, such that for each interval $I\subset\left[ 0,1\right] $ there is an idempotent $f$ so that $\int _{I}\left\vert f\left( x\right) \right\vert ^{p}dx\diagup\int_{0} ^{1}\left\vert f\left( x\right) \right\vert ^{p}dx>a$. We will explain how to produce such concentration for each $p>0$. The origin of this question and the history of the development of its solution will be surveyed.

On the Lattice Generated by Hamel Functions

Mon, 14 Mar 2011 09:08 EDT

Grzegorz Matusik

Source: Real Anal. Exchange, Volume 36, Number 1, 65--78.

Abstract:
We say that $f: \mathbb{R}\to \mathbb{R}$ is \lif\ if it is linearly independent over $\mathbb{Q}$ as a subset of $\mathbb{R}^2$ and that it is a Hamel function (\ham) if it is a Hamel basis of $\mathbb{R}^2$. In this paper we present a discussion on the lattices generated by the classes \ham\ and \lif. We also investigate extensions of partial \lif\ functions to \ham\ and \lif\ functions defined on whole $\mathbb{R}$.

Full Dimensional Sets Without Given Patterns

Mon, 14 Mar 2011 09:08 EDT

Péter Maga

Source: Real Anal. Exchange, Volume 36, Number 1, 79--90.

Abstract:
We construct a $d$ Hausdorff dimensional compact set in $\R^d$ that does not contain the vertices of any parallelogram. We also prove that for any given triangle ($3$ given points in the plane) there exists a compact set in $\R^2$ of Hausdorff dimension $2$ that does not contain any similar copy of the triangle. On the other hand, we show that the set of the $3$-point patterns of a $1$-dimensional compact set of $\R$ is dense.

Upper Porosity With Respect to Measures

Mon, 14 Mar 2011 09:08 EDT

Martin Koc

Source: Real Anal. Exchange, Volume 36, Number 1, 91--106.

Abstract:
For subsets of a separable metric space $X$ we introduce the notion of upper porosity with respect to a~Borel regular probabilistic measure $\mm$ on $X$ (called $\mm$-upper porosity) that generalizes the concept of upper porosity of the measure $\mm$. We explore several natural definitions and further provide a definition of even more general type of $\mm$-upper porosity given by suitable porosity functions. As the main consequence of achieved results concerning general $\mm$-upper porosities we get that every $\s$-$\mm$-upper porous set can be decomposed to a $\s$-strongly upper porous set and a $\mm$-null set.

A Feynman-Kac Solution to a Random Impulsive Equation of Schrödinger Type

Mon, 14 Mar 2011 09:08 EDT

E. M. Bonotto, M. Federson, P. Muldowney

Source: Real Anal. Exchange, Volume 36, Number 1, 107--148.

Abstract:
If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schr\"{o}\-din\-ger-type equation. We consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional ``impulsive'' displacement. Using a Feynman-Kac formulation based on generalized Riemann integration, we examine the resulting equation of motion.

Extension of continuous functions to Baire-one functions

Mon, 14 Mar 2011 09:08 EDT

Olena Karlova

Source: Real Anal. Exchange, Volume 36, Number 1, 149--160.

Abstract:
We introduce the notion of $B_1$-retract and investigate the connection between $B_1$- and $H_1$-retracts.

The Distribution Function and Measure Preserving Maps

Mon, 14 Mar 2011 09:08 EDT

Behrouz Emamizadeh

Source: Real Anal. Exchange, Volume 36, Number 1, 161--168.

Abstract:
Existence of measure preserving maps has been discussed in books on real analysis where the Axiom of Choice is instrumental. In this note we introduce a method to {\em construct} such maps. For our construction we use the distribution function and elementary differential equations.

Submultiplicativity of Norms for Spaces of Generalized BV-functions

Mon, 14 Mar 2011 09:08 EDT

Robert Kantrowitz

Source: Real Anal. Exchange, Volume 36, Number 1, 169--176.

Abstract:
The purpose of this article is to offer a couple of short arguments for results describing the interaction between the norm and pointwise products in certain spaces of functions of generalized bounded variation.

ℑᴷ-convergence

Mon, 14 Mar 2011 09:08 EDT

Martin Mačaj, Martin Sleziak

Source: Real Anal. Exchange, Volume 36, Number 1, 177--194.

Abstract:
In this paper we introduce $\IhJ$-convergence which is a common generalization of the $\I^*$-convergence of sequences, double sequences, and nets. We show that many results that were shown before for these special cases are true for the $\IhJ$-convergence, too.

Ultimately Increasing Functions

Mon, 14 Mar 2011 09:08 EDT

Gerald Beer, Jesús Rodríguez-López

Source: Real Anal. Exchange, Volume 36, Number 1, 195--212.

Abstract:
A function $g$ between directed sets $\langle \Sigma, \succeq' \rangle$ and $\langle \Lambda, \succeq \rangle$ is called \emph{ultimately increasing} if for each $\sigma_1 \in \Sigma$ there exists $\sigma_2 \succeq' \sigma_1$ such that $\sigma \succeq' \sigma_2 \Rightarrow g(\sigma)\succeq g(\sigma_1)$. A subnet of a net $a$ defined on $\langle \Lambda, \succeq \rangle$ \cite {Ke} is nothing but a composition of the form $a \circ g$ where $g$ is ultimately increasing and $g(\Sigma)$ is a cofinal subset of $\Lambda$. While even for linearly ordered sets, an increasing net defined on a cofinal subset of the domain need not have an increasing extension, in complete generality, it must have an ultimately increasing extension, and conversely when the domain is linearly ordered. Applications are given in the context of functions with values in a linearly ordered set equipped with the order topology - in particular, the extended real numbers. For example, we show that a real sequence $\langle a_n \rangle$ converges to the supremum of its set of terms if and only if $\langle a_n \rangle$ is the supremum of the ultimately increasing sequences that it majorizes.

On Inhomogeneous Bernoulli Convolutions and Random Power Series

Mon, 14 Mar 2011 09:08 EDT

Antonios Bisbas, Jörg Neunhäuserer

Source: Real Anal. Exchange, Volume 36, Number 1, 213--222.

Abstract:
We extend the results of Peres and Solomyak on absolute continuity and singularity of homogeneous Bernoulli convolutions to inhomogeneous ones and generalize the result to random power series given by inhomogeneous Markov chains. In addition we prove an Erd\"os-Salem type theorem for inhomogeneous Bernoulli convolutions.

There are Measurable Hamel Functions

Mon, 14 Mar 2011 09:08 EDT

Rafał Filipów, Andrzej Nowik, Piotr Szuca

Source: Real Anal. Exchange, Volume 36, Number 1, 223--230.

Abstract:
We say that a function $f : \real\to\real$ is a {\it Hamel function} if $f$, considered as a subset of $\real^2$, is a Hamel basis of $\real^2$. We show that there is a Marczewski measurable Hamel function. Additionally, we show that there is a Hamel function which is both Lebesgue measurable and with the Baire property.

An Example of a Quasi-continuous Hamel Function

Mon, 14 Mar 2011 09:08 EDT

Tomasz Natkaniec

Source: Real Anal. Exchange, Volume 36, Number 1, 231--236.

Abstract:
We say that $f: \mathR\to\mathR$ is a Hamel function if $f$, considered as a subset of $\mathR^2$, is a Hamel basis of $\mathR^2$. For a Cantor set $C\subset\mathR$ we construct a quasi-continuous Hamel function such that $f\restr(\mathR\setminus C)$ is of Baire class one.

On the Haar Measures in Topological Fields

Mon, 14 Mar 2011 09:08 EDT

Robert E. Zink

Source: Real Anal. Exchange, Volume 36, Number 1, 237--242.

Abstract:
By virtue of the uniqueness theorem for the Haar measure on a topological group, a simple argument is sufficient to show that the Haar measures on a locally--compact topological field, corresponding to the additive and multiplicative structures of the field, are absolutely continuous with respect to one another.

Wavelet Sets Accumulating at the Origin

Mon, 14 Mar 2011 09:08 EDT

Aparna Vyas, Rajeshwari Dubey

Source: Real Anal. Exchange, Volume 36, Number 1, 243--244.

Abstract:
This note corrects three typographical errors in the paper {\it Wavelet Sets Accumulating at the Origin}, Real Anal. Exchange {\bf 35(2)}, 463-478. The corrections are listed according to the page of the original article.

Duality between Measure and Category in Uncountable Locally Compact Abelian Polish Groups

Fri, 11 Nov 2011 09:08 EST

Richárd Balka

Source: Real Anal. Exchange, Volume 36, Number 2, 245--256.

Abstract:
We show that there is no addition preserving ErdŐs-Sierpiński mapping on any uncountable locally compact abelian Polish group. This generalizes results of Bartoszyński and Kysiak.

Topologies Generated by the $\psi$-Sparse Sets

Fri, 11 Nov 2011 09:08 EST

Anna Gożdziewicz-Smejda, Ewa Łazarow

Source: Real Anal. Exchange, Volume 36, Number 2, 257--268.

Abstract:
We study the notion of $\psi$-sparse point and $\psi$-sparse topology for nondecreasing continuous function $\psi$. We show that $\psi$-sparse topology is stronger then the $\psi$-density topology and weaker than the density topology.

Products of Baire One Double Star Functions

Fri, 11 Nov 2011 09:08 EST

Agnieszka Łukasiewicz, Aleksander Maliszewski

Source: Real Anal. Exchange, Volume 36, Number 2, 269--296.

Abstract:
We characterize products of Baire one double star functions.

A Factorization Problem

Fri, 11 Nov 2011 09:08 EST

Jack Grahl, Togo Nishiura

Source: Real Anal. Exchange, Volume 36, Number 2, 297--306.

Abstract:
A solution is presented of a problem proposed at the Summer Symposium in Real Analysis XXXIII.

Maximal Classes for the Family of $[\lambda,\varrho]$-Continuous Functions

Fri, 11 Nov 2011 09:08 EST

Stanisław Kowalczyk, Katarzyna Nowakowska

Source: Real Anal. Exchange, Volume 36, Number 2, 307--324.

Abstract:
In this paper we give the definition of $[\lambda,\varrho]$-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of $[\lambda,\varrho]$-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of $[\lambda,\varrho]$-continuous functions.

On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

Fri, 11 Nov 2011 09:08 EST

Gogi Pantsulaia

Source: Real Anal. Exchange, Volume 36, Number 2, 325--340.

Abstract:
The concepts of uniformly distributed sequences of an increasing family of finite sets and Riemann integrability are considered in terms of the “Lebesgue measure” on infinite-dimensional rectangles in $R^{\infty}$ and infinite-dimensional versions of famous results of Lebesgue and Weyl are proved.

On the Comparison of Density Type Topologies Generated by Functions

Fri, 11 Nov 2011 09:08 EST

Małgorzata Filipczak, Tomasz Filipczak

Source: Real Anal. Exchange, Volume 36, Number 2, 341--352.

Abstract:
In the paper there is presented a necessary and sufficient condition to compare $f$-density topologies.

On Absolute Convergence of Fourier Integrals

Fri, 11 Nov 2011 09:08 EST

E. Liflyand

Source: Real Anal. Exchange, Volume 36, Number 2, 353--360.

Abstract:
New sufficient conditions for representation of a function as an absolutely convergent Fourier integral are obtained in the paper.

Learning Theorems

Fri, 11 Nov 2011 09:08 EST

Jan Mycielski

Source: Real Anal. Exchange, Volume 36, Number 2, 361--372.

Abstract:
We will prove learning theorems that could explain, if only a little, how some organisms generalize information that they get from their senses.

Triangle Integral -- A Nonabsolute Integration Process Suitable for Piecewise Linear Surfaces

Fri, 11 Nov 2011 09:08 EST

Pedro L. Kaufmann, Ricardo Bianconi

Source: Real Anal. Exchange, Volume 36, Number 2, 373--404.

Abstract:
We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the $M_1$-integral presented in \cite{jks}, using triangle-based partitions.

Some Estimates of Commutators

Fri, 11 Nov 2011 09:08 EST

Chunping Xie

Source: Real Anal. Exchange, Volume 36, Number 2, 405--416.

Abstract:
By using the boundedness of the maximal and sharp operators on Morrey spaces, we have proved that the commutators $[M_p,b]$ and $[M^{\#},b]$ are bounded on Morrey spaces $L^{q,\lambda}$ if and only if $b$ is in BMO and the negative part of $b$ is in $L^{\infty}$.

Integral Representations for a Class of Operators on $L_{E}^{1}$

Fri, 11 Nov 2011 09:08 EST

Surjit S. Khurana

Source: Real Anal. Exchange, Volume 36, Number 2, 417--420.

Abstract:
Let $(X, \mathcal{A}, \mu)$ be a finite measure space, $E$ a locally convex Hausdorff space, $L_{E}^{1}$ the space of functions $f: X \to E$ which are $\mu$-integrable by semi-norms, $P(\mu, E)$ the space of Pettis integrable functions and $P_{1}(\mu, E)$ those elements of $P(\mu, E)$ which are measurable by semi-norms. We prove that a linear continuous mapping $ T: L_{E}^{1} \to E$ is of the form $T(f)= \int g f d \mu$ ($g \in L^{\infty}$) if and only if $ h( T(f))=0$ whenever $ h\circ f=0 $ for any $ f \in L_{E}^{1}, h \in E'$. Similar results are proved for $P(\mu, E)$ and $P_{1}(\mu, E)$.

Convergence of Automorphisms and Semicontinuity of Automorphism Groups

Fri, 11 Nov 2011 09:08 EST

Steven G. Krantz

Source: Real Anal. Exchange, Volume 36, Number 2, 421--434.

Abstract:
We study the compactness of the automorphism group of a domain in $\CC^n$, and in particular the convergence properties of mappings. We supply an application to the semicontinuity of automorphism groups under perturbation of the underlying domain. Relevant examples are provided.

Henstock Type Integral for Vector Valued Functions in a Compact Metric Space

Fri, 11 Nov 2011 09:08 EST

Caterina La Russa

Source: Real Anal. Exchange, Volume 36, Number 2, 435--448.

Abstract:
We define a Henstock-type integral for vector valued functions defined in a probability metric compact Radon space, using a suitable family ${\mathcal B}$ of measurable sets which play the role of ``\textit{intervals}''. When ${\mathcal B}$ is the family of all subintervals of $[0,1]$ we obtain the classical Henstock-Kurzweil integral on the real line, whereas if ${\mathcal B}$ is the family of all subintervals of $[0,1]^2$, or that of all subintervals of $[0,1]^2$ with a fixed regularity, we obtain the classical Henstock integral on the plane with respect to the Kurzweil base or the Kempisty base respectively.

Irreducibility, Infinite Level Sets and Small Entropy

Fri, 11 Nov 2011 09:08 EST

Jozef Bobok, Martin Soukenka

Source: Real Anal. Exchange, Volume 36, Number 2, 449--462.

Abstract:
We investigate continuous piecewise affine interval maps with countably many laps that preserve the Lebesgue measure. In particular, we construct such maps having knot points (a point $x$ where Dini's derivatives satisfy $D^{+}f(x)=D^{-}f(x)= \infty$ and $D_{+}f(x)=D_{-}f(x)= -\infty$) and estimate their topological entropy. Our main result is: for any $\varepsilon>0$ we construct a continuous interval map $g=g_{\varepsilon}$ such that (i) $g$ preserves the Lebesgue measure; (ii) knot points of $g$ are dense in $[0,1]$ and for a $G_{\delta}$ dense set of $z$'s, the set $g^{-1}(\{z\})$ is infinite; (iii) $\h_{\text{top}}(g)\le\log2+\varepsilon$.

On the Continuous Functions with Respect to $\mathcal{I} (J)$-Density Topologies

Fri, 11 Nov 2011 09:08 EST

Jacek Hejduk, Renata Wiertelak

Source: Real Anal. Exchange, Volume 36, Number 2, 463--470.

Abstract:
This paper contains the properties of continuous functions equipped with the $\mathcal{I}(J)$--density topology or natural topology in the domain or the range.

On Section Sets of Neighborhoods of Graphs of Semicontinuous Functions

Fri, 11 Nov 2011 09:08 EST

Dušan Pokorný

Source: Real Anal. Exchange, Volume 36, Number 2, 471--478.

Abstract:
We prove that for any lower semicontinuous function $f:[0,1]\to[0,1]$ with purely unrectifiable graph and for any $\varepsilon>0$ there is an open set $U\supset \graph f$\, with every vertical section set of one-dimensional Lebesgue measure at most $\varepsilon.$

Another Proof That $L^{p}$-Bounded Pointwise Convergence Implies Weak Convergence

Fri, 11 Nov 2011 09:08 EST

Marian Jakszto

Source: Real Anal. Exchange, Volume 36, Number 2, 479--482.

Abstract:
This note gives another proof of the known fact that $L^{p}$-bounded pointwise convergence implies weak convergence in $L^{p},$ $p>1.$ The proof is based on Banach and Saks’ theorem. The same method applies to convergence in measure.

Glaeser's Inequality on an Interval

Fri, 11 Nov 2011 09:08 EST

Adam Coffman, Yifei Pan

Source: Real Anal. Exchange, Volume 36, Number 2, 483--490.

Abstract:
We use elementary methods to find pointwise bounds for the first derivative of a real valued function with a continuous, bounded second derivative on an interval.

A Counterexample for the Change of Variable Formula in KH Integrals

Fri, 11 Nov 2011 09:08 EST

Michael Bensimhoun

Source: Real Anal. Exchange, Volume 36, Number 2, 491--498.

Abstract:
It was shown in \cite{B} and \cite{Ke} that if $F(x)$ and $\Psi(x)$ are Riemann integrals of the form $\int_a^x f\dx$ and $\int_b^x \psi \dx$ resp., then $\psi\cdot f\circ\Psi$, if defined, is Riemann integrable. Furthermore, the change of variable formula applies, giving $\int_b^x \psi\cdot f\circ\Psi \, \dx = F(\psi(x))-F(\psi(b))$. It is natural to try to generalize this theorem to the Kurzweil-Henstock integral (this question was also dealt with in a paper by the author \cite{Be}); in other words, assuming that $F$ and $\Psi$ are KH integrals of $f$ and $\psi$ resp., one would expect that $\psi\cdot f\circ\Psi$ be KH integrable. We show in this paper that this is false, and produce a counterexample based on the middle-third Cantor set and some rudiments of fractal geometry. In other words, by a well known theorem, we prove that the composition of two ACG functions needs not be ACG (in fact, we prove more generally that the composition of two absolutely continuous functions needs not be ACG). Of course, examples that show that the composition of two absolutely continuous functions needs not be absolutely continuous exist in the context of the Lebesgue integral, but since KH integrals need not be absolutely continuous, one cannot infer from these examples the validity of the above claim in the context of KH integration. On the other hand, the subtle method developed in this paper seems to be new, is entirely constructive, and we believe it could be applied to other interesting constructions. \vskip 0pt plus 10pt

Characterizations of Some Subclasses of the First Class of Baire

Fri, 11 Nov 2011 09:08 EST

Monika Lindner, Sebastian Lindner

Source: Real Anal. Exchange, Volume 36, Number 2, 499--506.

Abstract:
In the paper \cite{3Ch} the authors have examined functions of the Baire class 1, where the domain and the range were metric spaces. The $\varepsilon - \delta$ characterization of such functions has been proved. In this note we examine, if replacing of the condition from \cite{3Ch} by it's stronger version can lead us to the characterization of some subclass of $B_1$ on the interval $[0,1]$.

The Takagi Function: a Survey

Mon, 30 Apr 2012 13:26 EDT

Pieter C. Allaart, Kiko Kawamura

Source: Real Anal. Exchange, Volume 37, Number 1, 1--54.

Abstract:
This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.

A Covering Theorem and the Random-Indestructibility of the Density Zero Ideal

Mon, 30 Apr 2012 13:26 EDT

Márton Elekes

Source: Real Anal. Exchange, Volume 37, Number 1, 55--60.

Abstract:
The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\si$-finite measure space $(X, \iA, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, then there exists a sequence of integers $n_i$ of density zero so that $A_{n_i}$ still covers $\mu$-a.e. $x \in X$ infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas.

Binomial Measures and their Approximations

Mon, 30 Apr 2012 13:26 EDT

Francesco Calabrò, Antonio Corbo Esposito, Carmen Perugia

Source: Real Anal. Exchange, Volume 37, Number 1, 61--82.

Abstract:
In this paper we consider the properties of a family of probability (continuous and singular) measures, which will be called Binomial measures because of their relationship with the binomial model in probability. These measures arise in many applications with different notations. Many properties in common with Lebesgue measure hold true for this family, sometimes unexpectedly.

Resonances for Graph Directed Markov Systems

Mon, 30 Apr 2012 13:26 EDT

Martial R. Hille

Source: Real Anal. Exchange, Volume 37, Number 1, 83--116.

Abstract:
In this paper we introduce and study a certain zeta function and its zeros for conformal graph directed Markov systems (GDMS). These zeros are referred to as resonances. We specify a list of geometric, combinatoric and analytic conditions on the GDMS under which this zeta function is indeed well defined and even holomorphic on the whole complex plane. In addition, we prove that there is a half-plane where there are no zeros. Finally, we transfer a result of Guillopé {\it et al.} in \cite{GuLiZw} on the zeros of the Selberg zeta function to our setting. More precisely, we give an upper bound for the number of resonances in a strip in terms of the Hausdorff dimension of the limit set of the GDMS. We also briefly discuss relations to other zeta functions, in particular to the Selberg zeta function associated to a Kleinian group of Schottky type and to the geometric zeta function associated to a fractal string. Since the definition of the zeta function introduced in our paper is based on the transfer operator associated to the GDMS, these relations to other zeta functions indicate that our zeta function is a natural generalization of these zeta functions to conformal GDMSs.

Box Dimension of the Limit of Hölder Functions

Mon, 30 Apr 2012 13:26 EDT

Loredana Biacino

Source: Real Anal. Exchange, Volume 37, Number 1, 117--128.

Abstract:
In this note a theorem to determine the box dimension of the graph of the limit of a sequence of α-Hölder functions is established. By application of such a theorem the box dimensions of the graphs of some functions that are generalizations of Weierstrass-type functions are determined.

Moments and the Range of the Derivative

Mon, 30 Apr 2012 13:26 EDT

Eugen J. Ionascu, Richard Stephens

Source: Real Anal. Exchange, Volume 37, Number 1, 129--146.

Abstract:
In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first $n+1$ ($n\in \mathbb N\cup \{0\}$) moments, $\alpha_0$, $\alpha_1$,..., $\alpha_n$, of a real-valued continuously differentiable function $f$ defined on $[0,1]$, what can be said about the size of the image of $\frac{df}{dx}$? We make the questions more precise and we give answers in the cases of three or fewer moments and in some cases for four moments. In the general situation of $n+1$ moments, we show that the range of the derivative should contain the convex hull of a set of $n$ numbers calculated in terms of the Bernstein polynomials, $x^k(1-x)^{n+1-k}$, $k=1,2,...,n$, which turn out to involve expressions just in terms of the given moments $\alpha_i$, $i=0,1,2,...n$. In the end we make some conjectures about what may be true in terms of the sharpness of the interval range mentioned before.

Variation-Diminishing Wavelets and Wavelet Transforms

Mon, 30 Apr 2012 13:26 EDT

R. S. Pathak

Source: Real Anal. Exchange, Volume 37, Number 1, 147--166.

Abstract:
Using Schoenberg's theory of variation-diminishing integral operators of convolution type variation diminishing wavelets and wavelets of specific changes in sign are constructed. An inversion formula involving derivatives of the wavelet transform is established. Wavelets generated by Tanno's form of convolution kernels and $H$-functions are also investigated. Results are illustrated by means of examples and figures.

Remarks on the Continuity of Functions of Two Variables

Mon, 30 Apr 2012 13:26 EDT

Michael McAsey, Libin Mou

Source: Real Anal. Exchange, Volume 37, Number 1, 167--176.

Abstract:
The continuity of $f(x,y)$ at $(x_0,y_0)$ can be described by the behavior of $f$ along a collection of paths toward $(x_0,y_0)$ if the collection is rich enough. The collection of paths that are $\mathcal{C}^1$ and convex is rich enough but the collection of differentiable functions with bounded derivatives is not. The collection of $\mathcal{C}^n$ parameterized paths $(x(t),y(t))$ for any $n\gt 0$ is also rich enough to capture continuity.

BV p -Functions and Change of Variable

Mon, 30 Apr 2012 13:26 EDT

N. Merentes, J. L. Sánchez

Source: Real Anal. Exchange, Volume 37, Number 1, 177--188.

Abstract:
In this note we discuss some interconnections between the space $BV_p[a,b]$ ($1\leq p\lt\infty$) of functions of bounded $p$-variation (in Wiener's sense) and the space $Lip_\alpha[a,b]$ ($0\lt\alpha\leq 1$) of Hölder continuous functions. In particular, we show that $f\in BV_p[a,b]$ if and only if $f=g\circ \tau$, with $g\in Lip_{1/p}[a,b]$ and $\tau$ being monotone, and that $f\in BV_p[a,b] \cap C[a,b]$ if and only if $f=g\circ \tau$, with $g\in Lip_{1/p}[a,b]$ and $\tau$ being a homeomorphism.

Least Squares and Approximate Differentiation

Mon, 30 Apr 2012 13:26 EDT

Russell A. Gordon

Source: Real Anal. Exchange, Volume 37, Number 1, 189--202.

Abstract:
The least squares derivative and the approximate derivative are both generalizations of the ordinary derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other and it is even possible for both generalized derivatives to exist at a point but have different values. Several examples of such functions are presented in this paper. In addition, conditions for which the existence of the approximate derivative implies the existence (and equality) of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability.

On an Example of a Function with a Derivative which does not have a Third Order Symmetric Riemann Derivative Anywhere

Mon, 30 Apr 2012 13:26 EDT

John C. Georgiou

Source: Real Anal. Exchange, Volume 37, Number 1, 203--212.

Abstract:
In this paper we construct a differentiable function $ F : \mathbb{R} \to \mathbb{R} $ that does not have a third order symmetric Riemann derivative at any point. In fact, \[ \underline{SRD}^3F(x) = \liminf_{h \to 0} \tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = - \infty \] and \[ \overline{SRD}^3F(x) = \limsup_{h \to 0 }\tfrac{F(x + 3h) - 3F(x+h) + 3F(x - h) - F(x - 3h)}{(2h)^3} = + \infty \] for every $ x \in \mathbb{R}. $

Uniform Continuity of a Product of Real Functions

Mon, 30 Apr 2012 13:26 EDT

G. Beer, S. Naimpally

Source: Real Anal. Exchange, Volume 37, Number 1, 213--220.

Abstract:
We produce necessary and sufficient conditions for the pointwise product of two uniformly continuous real-valued functions defined on a metric space to be uniformly continuous.

On Riemann Sums

Mon, 30 Apr 2012 13:26 EDT

Brian S. Thomson

Source: Real Anal. Exchange, Volume 37, Number 1, 221--242.

Abstract:
If a sum of the form \[\sum_{i=1}^n f(\xi_i)(x_i-x_{i-1})\] is used without the familiar requirement that the sequence of points $a=x_0$, $x_1$, \dots, $x_n=b$ is increasing, do we still get a useful approximation to the integral? With a suitable set of hypotheses the answer is yes. We give applications to change of variable formulas and the problem of characterizing derivatives.

Points of Middle Density in the Real Line

Mon, 15 Apr 2013 08:57 EDT

Marianna Csörnyei, Jack Grahl, Toby C. O'Neil

Source: Real Anal. Exchange, Volume 37, Number 2, 243--248.

Abstract:
A Lebesgue measurable set in the real line has Lebesgue density 0 or 1 at almost every point. Kolyada showed that there is a positive constant $\delta$ such that for non-trivial measurable sets there is at least one point with upper and lower densities lying in the interval $(\delta, 1-\delta)$. Both Kolyada and later Szenes gave bounds for the largest possible value of this $\delta$. In this note we reduce the best known upper bound, disproving a conjecture of Szenes.

Limit Theorems in ( l )-Groups with Respect to ( D )-Convergence

Mon, 15 Apr 2013 08:57 EDT

Antonio Boccuto, Xenofon Dimitriou, Nikolaos Papanastassiou

Source: Real Anal. Exchange, Volume 37, Number 2, 249--278.

Abstract:
Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for $(l)$-group-valued measures are given in the context of $(D)$-convergence. We consider both the $\sigma$-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measures is intended relatively to a common $(D)$-sequence. Among the tools, we use the Fremlin lemma, which allows us to replace a countable family of $(D)$-sequence with one regulator, and the Maeda-Ogasawara-Vulikh representation theorem for Archimedean lattice groups.

On Laplace Continuity

Mon, 15 Apr 2013 08:57 EDT

S. Ray, T. K. Garai

Source: Real Anal. Exchange, Volume 37, Number 2, 279--290.

Abstract:
Some properties of Laplace continuous functions and Laplace derivable functions are studied.

Strictly Monotone Functions on Preimages of Open Sets Leading to Lyapunov Functions

Mon, 15 Apr 2013 08:57 EDT

Dan Dobrovolschi

Source: Real Anal. Exchange, Volume 37, Number 2, 291--304.

Abstract:
We consider a real function $f$ of a real variable such that, for every point $x$ of the preimage $f^{-1}(D)$ of a set $D \subseteq \mathbb{R}$, $f$ is strictly monotone at $x$, and give sufficient conditions of strict monotonicity of $f$ on $f^{-1}(D)$. In particular, we prove that a differentiable function $f$ on an open interval, whose derivative is strictly negative on $f^{-1}(D)$, where $D \subseteq \mathbb{R}$ is an open set, is strictly decreasing on $f^{-1}(D)$. The latter result has applications in stability theory of differential equations on $\mathbb{R}^N$. The first application provides Lyapunov functions $V$ for preimages under $V$ of closed sets. The second application is a generalization of the Lyapunov stability theorem, in which the role of the asymptotically equilibrium point is played by $V^{-1}(-\infty, c_0]$, where $V$ is a Lyapunov function for $V^{-1}(-\infty, c_0]$, and all sublevel sets of $V$ are assumed to be compact. Moreover, due to compactness, all solutions of the differential equation are global to the right. The second application is also a generalization of a boundedness result from Geophysical Fluid Dynamics; in particular, it proves rigorously that all trajectories of the famous Lorenz system eventually enter a compact set.

Continuous and Smooth Images of Sets

Mon, 15 Apr 2013 08:57 EDT

Krzysztof Chris Ciesielski, Togo Nishiura

Source: Real Anal. Exchange, Volume 37, Number 2, 305--314.

Abstract:
This note shows that if a subset $S$ of $\real$ is such that some continuous function $f\colon\real\to\real$ has the property ``$f[S]$ contains a perfect set,'' then some $\C^\infty$ function $g\colon\real\to\real$ has the same property. Moreover, if $f[S]$ is nowhere dense, then the $g$ can have the stronger property ``$g[S]$ is perfect.'' The last result is used to show that it is consistent with ZFC (the usual axioms of set theory) that for each subset $S$ of $\real$ of cardinality $\continuum$ (the cardinality of the continuum) there exists a $\C^\infty$ function $g\colon \real\to\real$ such that $g[S]$ contains a perfect set.

A Note on Comparisons Between Birkhoff and McShane-Type Integrals for Multifunctions

Mon, 15 Apr 2013 08:57 EDT

Antonio Boccuto, Anna Rita Sambucini

Source: Real Anal. Exchange, Volume 37, Number 2, 315--324.

Abstract:
Here we present some comparison results between Birkhoff and McShane multivalued integration.

The Structure of Arithmetic Sums of Affine Cantor Sets

Mon, 15 Apr 2013 08:57 EDT

Razvan Anisca, Christopher Chlebovec, Monica Ilie

Source: Real Anal. Exchange, Volume 37, Number 2, 325--332.

Abstract:
In this paper we describe the structure of the arithmetic sum of two affine Cantor sets. These are self-similar sets which are part of the dynamically defined Cantor sets. Let ${\bf C_1}, {\bf C_2}$ be affine Cantor sets with $[0,s]$ and $[0,r]$ as intervals of step 0. We explicit a generic family of these self-similar sets for which the structure of ${\bf C_1}+{\bf C_2}$ is of one of the following five types: {\bf (i)} an $M$-Cantorval, {\bf (ii)} an $R$-Cantorval, {\bf (iii)} an $L$-Cantorval, or there exist $\lambda, \eta >0$ and intervals $I$, ${\tilde I}$ of the construction of ${\bf C_1}$ and ${\bf C_2}$, respectively, such that {\bf (iv)} $\lambda {\bf C_1}+\eta {\bf C_2}= {\bf C_1}\cap I + {\bf C_2}\cap {\tilde I} -\min I - \min {\tilde I} \,= \,[0, \lambda s + \eta r]$,\, or \,{\bf (v)} $\lambda {\bf C_1}+\eta {\bf C_2}= {\bf C_1}\cap I + {\bf C_2}\cap {\tilde I} -\min I - \min {\tilde I}$ is homeomorphic to the Cantor ternary set. This result generalizes the description obtained by Mendes and Oliveira for the case of homogeneous Cantor sets and the one obtained by the first two authors for semi-homogeneous Cantor sets. It also provides a suitable framework in which a question of Mendes and Oliveira admits a positive answer.

The Hausdorff Dimension of Graphs of Prevalent Continuous Functions

Mon, 15 Apr 2013 08:57 EDT

Jonathan M. Fraser, James T. Hyde

Source: Real Anal. Exchange, Volume 37, Number 2, 333--352.

Abstract:
We prove that the Hausdorff dimension of the graph of a prevalent continuous function is 2. We also indicate how our results can be extended to the space of continuous functions on $[0,1]^d$ for $d \in \mathbb{N}$ and use this to obtain results on the `horizon problem' for fractal surfaces. We begin with a survey of previous results on the dimension of a generic continuous function.

Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

Mon, 15 Apr 2013 08:57 EDT

Krzysztof Chris Ciesielski, Timothy Glatzer

Source: Real Anal. Exchange, Volume 37, Number 2, 353--362.

Abstract:
We provide a simple construction of a function $F\colon\real^2\to\real$ discontinuous on a perfect set $P$, while having continuous restrictions $F\restriction C$ for all twice differentiable curves $C$. In particular, $F$ is separately continuous and linearly continuous. While it has been known that the projection $\pi[P]$ of any such set $P$ onto a straight line must be meager, our construction allows $\pi[P]$ to have arbitrarily large measure. In particular, $P$ can have arbitrarily large $1$-Hausdorff measure, which is the best possible result in this direction, since any such $P$ has Hausdorff dimension at most 1.

Mycielski-Regular Measures

Mon, 15 Apr 2013 08:57 EDT

Jeremiah J. Bass

Source: Real Anal. Exchange, Volume 37, Number 2, 363--374.

Abstract:
Let $\mu$ be a Radon probability measure on the Euclidean space $\mathbb{R}^{d}$ for $d\geq 1$, and $f\:mathbb{R}^{d}\to \mathbb{R}$ a measurable function. Given a sequence in $(\mathbb{R}^{d})^{\mathbb{N}}$, for any $x\in\mathbb{R}^{d}$ define $f_{n}(x)=f(x_{k})$, where $x_{k}$ is the first among $x_{0},\ldots, x_{n-1}$ that minimizes the distance from $x$ to $x_{k}$, $0 \leq k\leq n-1$. The measures for which the sequence $(f_{n})_{n=1}^{\infty}$ converges in measure to $f$ for almost every sequence $(x_{0},x_{1},\ldots)$ are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set $C$ is Mycielski-regular.

Dimension Spectrum for a Nonconventional Ergodic Average

Mon, 15 Apr 2013 08:57 EDT

Yuval Peres, Boris Solomyak

Source: Real Anal. Exchange, Volume 37, Number 2, 375--388.

Abstract:
We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of $0,1$ sequences, for which the frequency of the pattern 11 in positions $k, 2k$ equals a given number $\theta\in [0,1]$.

Ideal Exhaustiveness, Weak Convergence and Weak Compactness in Banach Spaces

Mon, 15 Apr 2013 08:57 EDT

Antonio Boccuto, Pratulananda Das, Xenofon Dimitriou, Nikolaos Papanastassiou

Source: Real Anal. Exchange, Volume 37, Number 2, 389--410.

Abstract:
Some types of compactness in the ideal context are defined and relations between ideal exhaustiveness and equicontinuity of measures are investigated. As applications, some versions of limit theorems involving ideal pointwise convergence of measure sequences and some weak compactness results related to integral functionals are presented.

The Itô-Henstock Stochastic Differential Equations

Mon, 15 Apr 2013 08:57 EDT

Tan Soon Boon, Toh Tin Lam

Source: Real Anal. Exchange, Volume 37, Number 2, 411--424.

Abstract:
In this paper, we study the stochastic integral equation with its stochastic integral defined using the Henstock approach, or commonly known as the generalized Riemann approach, instead of the classical Itô integral, which we shall call it the Itô-Henstock integral equation. Our aim is to prove the existence of solution of the Itô-Henstock integral equation using the well known method used in the existence theorem of the ordinary differential equation, namely the Picard's iteration method.

Uniform Continuity of a Product of Real Functions

Mon, 15 Apr 2013 08:57 EDT

Sanjib Basu

Source: Real Anal. Exchange, Volume 37, Number 2, 425--438.

Abstract:
To every Lebesgue measurable subset of $\mathbb{R}$ is associated a certain subcollection of points where the given measurable set possesses a density. By virtue of Lebesgue's famous theorem on metric density, this associated set is a set of full measure in $\mathbb{R}$ and is hence measure-theoretically very large. But are these sets also topologically large? In Lebesgue's theorem, the set is kept fixed while the point is allowed to vary. If instead, we keep the point fixed a vary the set, then we may have corresponding to each point in $\mathbb{R}$ a certain subclass of measurable sets each member of which possesses a density at that point. How large is this subclass in the ``topology of measurable subsets of $\mathbb{R}$"? In this paper, in an endeavour to seek out answers to the questions set above, we have arrived at certain interesting and significant conclusions. Somewhat similar conclusions have been derived over analogous questions relating to `set-porosity'.

Mean Value Integral Inequalities

Mon, 15 Apr 2013 08:57 EDT

Rodrigo López Pouso

Source: Real Anal. Exchange, Volume 37, Number 2, 439--450.

Abstract:
Let $F:[a,b]\longrightarrow \R$ have zero derivative in a dense subset of $[a,b]$. What else we need to conclude that $F$ is constant in $[a,b]$? We prove a result in this direction using some new Mean Value Theorems for integrals which are the real core of this paper. These Mean Value Theorems are proven easily and concisely using Lebesgue integration, but we also provide alternative and elementary proofs to some of them which keep inside the scope of the Riemann integral.

Uniform Differentiability

Mon, 15 Apr 2013 08:57 EDT

Julius V. Benitez, Ferdinand P. Jamil, Chew Tuan Seng

Source: Real Anal. Exchange, Volume 37, Number 2, 451--462.

Abstract:
The concept of uniform differentiability is introduced to characterize sequences of McShane and Henstock equi-integrable functions.

A New proof of the Sobczyk-Hammer Decomposition Theorem

Mon, 15 Apr 2013 08:57 EDT

Zafer Ercan

Source: Real Anal. Exchange, Volume 37, Number 2, 463--466.

Abstract:
In this short note, we give a relatively simple proof of the Łoś-Marczewski Extension of finitely additive measures. In particular, we extend the Łoś-Marczewski Extension to Dedekind complete Riesz-space-valued functions.

A Large Group of Nonmeasurable Additive Functions

Mon, 15 Apr 2013 08:57 EDT

Alexander B. Kharazishvili

Source: Real Anal. Exchange, Volume 37, Number 2, 467--476.

Abstract:
By assuming the Continuum Hypothesis, it is proved that there exists a subgroup of ${\bf R}^{{\bf R}}$ of cardinality strictly greater than the cardinality of the continuum, all nonzero members of which are absolutely nonmeasurable additive functions.

The Hakeʼs Theorem and Variational Measures

Mon, 15 Apr 2013 08:57 EDT

Surinder Pal Singh, Inder K. Rana

Source: Real Anal. Exchange, Volume 37, Number 2, 477--488.

Abstract:
We give a characterization of the Henstock-Kurzweil integral on $\R^m$ in terms of variational measures. As an application of this we prove a generalization of the Hake's theorem to $\R^m$.

A New Proof of the Sobczyk-Hammer Decomposition Theorem

Mon, 15 Apr 2013 08:57 EDT

Zafer Ercan

Source: Real Anal. Exchange, Volume 37, Number 2, 489--492.

Abstract:
In this short note, we give a simple proof of the Sobczyk-Hammer Decomposition Theorem in terms of Dedekind complete Riesz spaces.

Error Level Saturation for Popoff’s Generalized Derivative Operator

Mon, 15 Apr 2013 08:57 EDT

C. W. Groetsch

Source: Real Anal. Exchange, Volume 37, Number 2, 493--498.

Abstract:
A saturation result with respect to data error level is presented for an approximate derivative operator of Kyrille Popoff.

Corrigendum in: A Generalization of Density Topology and on Generalization of the Density Topology on the Real Line

Mon, 15 Apr 2013 08:57 EDT

Wojciech Wojdowski

Source: Real Anal. Exchange, Volume 37, Number 2, 499--502.

Abstract:
The notion of $\mathcal{A}_{d}-$density point introduced in \cite{wo1}\ leads to the operator $\Phi _{\mathcal{A}_{d}}\left( A\right) $ which is not a lower density operator. We present a counterexample and give a corrected definition which should be used in two previous papers to keep all results valid.

Real Analysis Exchange Articles (Project Euclid)

How to Concentrate Idempotents

On the Lattice Generated by Hamel Functions

Full Dimensional Sets Without Given Patterns

Upper Porosity With Respect to Measures

A Feynman-Kac Solution to a Random Impulsive Equation of Schrödinger Type

Extension of continuous functions to Baire-one functions

The Distribution Function and Measure Preserving Maps

Submultiplicativity of Norms for Spaces of Generalized BV-functions

ℑᴷ-convergence

Ultimately Increasing Functions

On Inhomogeneous Bernoulli Convolutions and Random Power Series

There are Measurable Hamel Functions

An Example of a Quasi-continuous Hamel Function

On the Haar Measures in Topological Fields

Wavelet Sets Accumulating at the Origin

Duality between Measure and Category in Uncountable Locally Compact Abelian Polish Groups

Topologies Generated by the \(\psi\)-Sparse Sets

Products of Baire One Double Star Functions

A Factorization Problem

Maximal Classes for the Family of \([\lambda,\varrho]\)-Continuous Functions

On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

On the Comparison of Density Type Topologies Generated by Functions

On Absolute Convergence of Fourier Integrals

Learning Theorems

Triangle Integral -- A Nonabsolute Integration Process Suitable for Piecewise Linear Surfaces

Some Estimates of Commutators

Integral Representations for a Class of Operators on \(L_{E}^{1}\)

Convergence of Automorphisms and Semicontinuity of Automorphism Groups

Henstock Type Integral for Vector Valued Functions in a Compact Metric Space

Irreducibility, Infinite Level Sets and Small Entropy

On the Continuous Functions with Respect to \(\mathcal{I} (J)\)-Density Topologies

On Section Sets of Neighborhoods of Graphs of Semicontinuous Functions

Another Proof That \(L^{p}\)-Bounded Pointwise Convergence Implies Weak Convergence

Glaeser's Inequality on an Interval

A Counterexample for the Change of Variable Formula in KH Integrals

Characterizations of Some Subclasses of the First Class of Baire

The Takagi Function: a Survey

A Covering Theorem and the Random-Indestructibility of the Density Zero Ideal

Binomial Measures and their Approximations

Resonances for Graph Directed Markov Systems

Box Dimension of the Limit of Hölder Functions

Moments and the Range of the Derivative

Variation-Diminishing Wavelets and Wavelet Transforms

Remarks on the Continuity of Functions of Two Variables

BV p -Functions and Change of Variable

Least Squares and Approximate Differentiation

On an Example of a Function with a Derivative which does not have a Third Order Symmetric Riemann Derivative Anywhere

Uniform Continuity of a Product of Real Functions

On Riemann Sums

Points of Middle Density in the Real Line

Limit Theorems in ( l )-Groups with Respect to ( D )-Convergence

On Laplace Continuity

Strictly Monotone Functions on Preimages of Open Sets Leading to Lyapunov Functions

Continuous and Smooth Images of Sets

A Note on Comparisons Between Birkhoff and McShane-Type Integrals for Multifunctions

The Structure of Arithmetic Sums of Affine Cantor Sets

The Hausdorff Dimension of Graphs of Prevalent Continuous Functions

Functions Continuous on Twice Differentiable Curves, Discontinuous on Large Sets

Mycielski-Regular Measures

Dimension Spectrum for a Nonconventional Ergodic Average

Ideal Exhaustiveness, Weak Convergence and Weak Compactness in Banach Spaces

The Itô-Henstock Stochastic Differential Equations

Uniform Continuity of a Product of Real Functions

Mean Value Integral Inequalities

Uniform Differentiability

A New proof of the Sobczyk-Hammer Decomposition Theorem

A Large Group of Nonmeasurable Additive Functions

The Hakeʼs Theorem and Variational Measures

A New Proof of the Sobczyk-Hammer Decomposition Theorem

Error Level Saturation for Popoff’s Generalized Derivative Operator

Corrigendum in: A Generalization of Density Topology and on Generalization of the Density Topology on the Real Line