Real Analysis Exchange Articles (Project Euclid)
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The latest articles from Real Analysis Exchange on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 14 Mar 2011 09:08 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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How to Concentrate Idempotents
http://projecteuclid.org/euclid.rae/1272376220
<strong>J. Marshall Ash</strong><p><strong>Source: </strong>Real Anal. Exchange, Volume 35, Number 1, 1--20.</p><p><strong>Abstract:</strong><br/> 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.
</p>projecteuclid.org/euclid.rae/1272376220_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTBowen’s Formula for Shift-Generated Finite Conformal Constructionshttp://projecteuclid.org/euclid.rae/1435759197<strong>Andrei E. Ghenciu</strong>, <strong>Mario Roy</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 99--112.</p><p><strong>Abstract:</strong><br/>
We study shift-generated finite conformal constructions; i.e. conformal constructions generated by a general shift (shift of finite type, sofic shift and non-sofic shift alike) over a finite alphabet. These constructions are not restricted to shifts of finite type or sofic shifts as in the classical limit set constructions. In particular, we prove that the limit sets of such constructions satisfy Bowen’s formula, which gives the Hausdorff dimension of the limit set as the zero of the topological pressure. We look at several examples, including a one-dimensional construction generated by the so-called context-free shift.
</p>projecteuclid.org/euclid.rae/1435759197_20150701100000Wed, 01 Jul 2015 10:00 EDTHausdorff and Packing Measures of Balanced Cantor Setshttp://projecteuclid.org/euclid.rae/1435759198<strong>Kathryn Hare</strong>, <strong>Ka-Shing Ng</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 113--128.</p><p><strong>Abstract:</strong><br/>
We estimate the \(h\)-Hausdorff and \(h\)-packing measures of balanced Cantor sets, and characterize the corresponding dimension partitions. This generalizes results known for Cantor sets associated with positive decreasing summable sequences and central Cantor sets.
</p>projecteuclid.org/euclid.rae/1435759198_20150701100000Wed, 01 Jul 2015 10:00 EDTOn Borel Hull Operationshttp://projecteuclid.org/euclid.rae/1435759199<strong>Tomasz Filipczak</strong>, <strong>Andrzej Rosłanowski</strong>, <strong>Saharon Shelah</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 129--140.</p><p><strong>Abstract:</strong><br/>
We show that some set-theoretic assumptions (for example Martin’s Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in \(\mathbb{R}^{n}\)). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \(\sigma\)-algebra of sets with the property of Baire.
</p>projecteuclid.org/euclid.rae/1435759199_20150701100000Wed, 01 Jul 2015 10:00 EDTGeneralized Kiesswetter’s Functionshttp://projecteuclid.org/euclid.rae/1435759200<strong>Delong Li</strong>, <strong>Jie Miao</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 141--156.</p><p><strong>Abstract:</strong><br/>
In 1966, Kiesswetter found an interesting example of continuous everywhere but differentiable nowhere functions using base-4 expansion of real numbers. In this paper we show how Kiesswetter’s function can be extended to general cases. We also provide an equivalent form for such functions via a recurrence relation.
</p>projecteuclid.org/euclid.rae/1435759200_20150701100000Wed, 01 Jul 2015 10:00 EDTAn Integral on a Complete Metric Measure Spacehttp://projecteuclid.org/euclid.rae/1435759201<strong>Donatella Bongiorno</strong>, <strong>Giuseppa Corrao</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 157--178.</p><p><strong>Abstract:</strong><br/>
We study a Henstock-Kurzweil type integral defined on a complete metric measure space \(X\) endowed with a Radon measure \(\mu\) and with a family of “cells” \(\mathcal{F}\) that satisfies the Vitali covering theorem with respect to \(\mu\). This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the \(s\)-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of \(ACG^*\) functions (Main Theorem 1) and in terms of
variational measures (Main Theorem 2).
</p>projecteuclid.org/euclid.rae/1435759201_20150701100000Wed, 01 Jul 2015 10:00 EDTA Sufficient Condition for a Bounded Set of Positive Lebesgue Measure in ℝ
2 or ℝ
3 to Contain its Centroidhttp://projecteuclid.org/euclid.rae/1435759202<strong>Eric A. Hintikka</strong>, <strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 179--192.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a sufficient condition for a domain in either two- or three-dimensional Euclidean space to contain its centroid. We show that the condition is sharp. The condition is not, however, necessary.
</p>projecteuclid.org/euclid.rae/1435759202_20150701100000Wed, 01 Jul 2015 10:00 EDTExtreme Results on Certain Generalized Riemann Derivativeshttp://projecteuclid.org/euclid.rae/1435759203<strong>John C. Georgiou</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 193--208.</p><p><strong>Abstract:</strong><br/>
In this paper the following question is investigated. Given a natural number \(r\) and numbers \(\alpha_j,\beta_j\) for \(j=0,1,\dots,r\) satisfying \( \alpha_0 <\alpha_1 < \dots <lt \alpha_r \) and \begin{equation*} \sum_{j=0}^{r} \beta_j \alpha_j^k= \begin{cases} 0 & \text{if \(k=0,1,\dots,r-1\)}\\ r!& \text{if \(k=r\) } \end{cases} \end{equation*} is there a \( 2\pi\)-periodic, \( r-1\) times continuously differentiable function \( f\) such that \begin{equation*} \limsup_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \limsup_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \infty, \end{equation*} \begin{equation*} \liminf_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \liminf_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = - \infty \end{equation*} for every \( x \in \mathbb{R} \)?
</p>projecteuclid.org/euclid.rae/1435759203_20150701100000Wed, 01 Jul 2015 10:00 EDTAbsolute Continuity in Partial Differential Equationshttp://projecteuclid.org/euclid.rae/1435759204<strong>Amin Farjudian</strong>, <strong>Behrouz Emamizadeh</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 209--218.</p><p><strong>Abstract:</strong><br/>
In this note, we study a function which frequently appears in partial differential equations. We prove that this function is absolutely continuous; hence it can be written as a definite integral. As a result, we obtain some estimates regarding solutions of the Hamilton-Jacobi systems.
</p>projecteuclid.org/euclid.rae/1435759204_20150701100000Wed, 01 Jul 2015 10:00 EDTBasic Introduction To Exponential and Logarithmic Functionshttp://projecteuclid.org/euclid.rae/1435759205<strong>Adel B. Badi</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 219--226.</p><p><strong>Abstract:</strong><br/>
This article discusses the definitions and properties of exponential and logarithmic functions. The treatment is based on the basic properties of real numbers, sequences and continuous functions. This treatment avoids the use of definite integrals.
</p>projecteuclid.org/euclid.rae/1435759205_20150701100000Wed, 01 Jul 2015 10:00 EDTAddendum to: Some new Types of Filter Limit Theorems for Topological group-valued Measureshttp://projecteuclid.org/euclid.rae/1435759206<strong>Antonio Boccuto</strong>, <strong>Xenofon Dimitriou</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 227--232.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to point out some corrections to the paper: A. Boccuto and X. Dimitriou, “Some new types of filter limit theorems for topological group-valued measures,” \textit{Real Anal. Exchange} \textbf{39} (1) (2014), 139-174.
</p>projecteuclid.org/euclid.rae/1435759206_20150701100000Wed, 01 Jul 2015 10:00 EDTOn Interval Based Generalizations of Absolute Continuity for Functions on \(\mathbb{R}^{n}\)http://projecteuclid.org/euclid.rae/1490580012<strong>Michael Dymond</strong>, <strong>Beata Randrianantoanina</strong>, <strong>Huaqiang Xu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 49--78.</p><p><strong>Abstract:</strong><br/> We study notions of absolute continuity for functions defined on $\mathbb{R}^n$ similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish some containment relations of the class $1\po AC_{\rm WDN}$ which consits of all functions in $1\po AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.
</p>projecteuclid.org/euclid.rae/1490580012_20170326220033Sun, 26 Mar 2017 22:00 EDTA Class of Random Cantor Setshttp://projecteuclid.org/euclid.rae/1490580013<strong>Changhao Chen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 79--120.</p><p><strong>Abstract:</strong><br/> In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.
</p>projecteuclid.org/euclid.rae/1490580013_20170326220033Sun, 26 Mar 2017 22:00 EDTVariance Jensen Type Inequalities for General Lebesgue Integral with Applicationshttp://projecteuclid.org/euclid.rae/1490580014<strong>Silvestru S. Dragomir</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 121--148.</p><p><strong>Abstract:</strong><br/> Some inequalities similar to Jensen inequalities for general Lebesgue integral are obtained. Applications for functions of selfadjoint operators and functions of unitary operators on complex Hilbert spaces are provided as well.
</p>projecteuclid.org/euclid.rae/1490580014_20170326220033Sun, 26 Mar 2017 22:00 EDTQuantization for Uniform Distributions on Equilateral Triangleshttp://projecteuclid.org/euclid.rae/1490580015<strong>Carl P. Dettmann</strong>, <strong>Mrinal Kanti Roychowdhury</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 149--166.</p><p><strong>Abstract:</strong><br/> We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries.
</p>projecteuclid.org/euclid.rae/1490580015_20170326220033Sun, 26 Mar 2017 22:00 EDTThe $\ell_1$-Dichotomy Theorem with Respect to a Coidealhttp://projecteuclid.org/euclid.rae/1490580016<strong>Vassiliki Farmaki</strong>, <strong>Andreas Mitropoulos</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 167--184.</p><p><strong>Abstract:</strong><br/> In this paper we introduce, for any coideal basis $\B$ on the set $\nat$ of natural numbers, the notions of a $\B$-sequence, a $\B$-subsequence of a $\B$-sequence, and a $\B$-convergent sequence in a metric space. The usual notions of a sequence, subsequence, and convergent sequence obtain for the coideal $\B$ of all the infinite subsets of $\nat$. We first prove a Bolzano-Weierstrass theorem for $\B$-sequences: if $\B$ is a Ramsey coideal basis on $\nat$, then every bounded $\B$-sequence of real numbers has a $\B$-convergent $\B$-subsequence; and next, with the help of this extended Bolzano-Weierstrass theorem, we establish an extension of the fundamental Rosenthal's $\ell_1$-dichotomy theorem: if $\B $ is a semiselective coideal basis on $\nat$, then every bounded $\B$-sequence of real valued functions $(f_n)_{n\in A}$ has a $\B$-subsequence $(f_n)_{n\in B}$, which is either $\B$-convergent or equivalent to the unit vector basis of $\ell_1(B)$.
</p>projecteuclid.org/euclid.rae/1490580016_20170326220033Sun, 26 Mar 2017 22:00 EDTDirectional Differentiability in the Euclidean Planehttp://projecteuclid.org/euclid.rae/1490580017<strong>J. Marshall Ash</strong>, <strong>Stefan Catoiu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 185--192.</p><p><strong>Abstract:</strong><br/> Smoothness conditions on a function $f\:mathbb{R}^{2}\rightarrow \mathbb{R}$ that are weaker than being differentiable or Lipschitz at a point are defined and studied.
</p>projecteuclid.org/euclid.rae/1490580017_20170326220033Sun, 26 Mar 2017 22:00 EDTBanach Spaces for the Schwartz Distributionshttps://projecteuclid.org/euclid.rae/1525226419<strong>Tepper L. Gill</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
This paper is a survey of a new family of Banach spaces \(\mcB\) that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the \(L^p\) spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions \(\mcD\) as a continuous dense embedding. Thus, by the Hahn-Banach theorem, \(\mcD' \subset \mcB'\). A new family that extend the space of functions of bounded mean oscillation \(BMO[\mathbb{R}^n]\), to include the HK-integrable functions are also introduced.
</p>projecteuclid.org/euclid.rae/1525226419_20180501220024Tue, 01 May 2018 22:00 EDTLinear Subspaces of Hypercyclic Vectorshttps://projecteuclid.org/euclid.rae/1525226420<strong>Juan Bês</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 37--42.</p><p><strong>Abstract:</strong><br/>
\noindent In my talk I presented results from previous papers on the existence of hypercyclic algebras for convolution operators acting on the space of entire functions.
</p>projecteuclid.org/euclid.rae/1525226420_20180501220024Tue, 01 May 2018 22:00 EDTSome Results about Big and Little Liphttps://projecteuclid.org/euclid.rae/1525226421<strong>Bruce Hanson</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 43--50.</p><p><strong>Abstract:</strong><br/>
Let \(f\:ℝ \to ℝ\) be continuous. We examine the relationship between the so-called “big Lip” and “little lip” functions: Lip \(f\) and lip \(f\).
</p>projecteuclid.org/euclid.rae/1525226421_20180501220024Tue, 01 May 2018 22:00 EDTMeasuring Anisotropy in Planar Setshttps://projecteuclid.org/euclid.rae/1525226422<strong>Toby C. O’Neil</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 51--56.</p><p><strong>Abstract:</strong><br/>
We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anistropy of fractal sets.
</p>projecteuclid.org/euclid.rae/1525226422_20180501220024Tue, 01 May 2018 22:00 EDTContinued Logarithm Representation of Real Numbershttps://projecteuclid.org/euclid.rae/1525226423<strong>Jörg Neunhäuserer</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 57--66.</p><p><strong>Abstract:</strong><br/>
We introduce the continued logarithm representation of real numbers and prove results on the occurrence and frequency of digits with respect to this representation.
</p>projecteuclid.org/euclid.rae/1525226423_20180501220024Tue, 01 May 2018 22:00 EDTAn Elementary Proof of an Isoperimetric Inequality for Paths with Finite p-Variationhttps://projecteuclid.org/euclid.rae/1525226424<strong>George Galvin</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 67--76.</p><p><strong>Abstract:</strong><br/>
In this article we will prove that if the continuous closed curve \(\gamma : [0, 1] \rightarrow \mathbb{R}^2\) has finite \(p\)-variation with \(p < 2\), then \begin{equation*} (\iint\limits_{\R^2}|\eta(\gamma, (x, y))|^q \,dx \,dy)^{1/q} \le (\frac{1}{2})^\frac{1}{q}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p, [0, 1]})^{\frac{2}{q}} \end{equation*} for all \(q \in [1, \frac{2}{p})\), where \(\eta(\gamma, (x, y))\) is the winding number of \(\gamma\) at \((x, y), \zeta\) is the Reimann zeta function, and \(||\gamma||_{p, [0, 1]}\) is the \(p\)-variation of \(\gamma\) on the interval \([0, 1]\). Our main contribution is that we have explicitly given a bound by known constants, and we have found this by an elementary proof. We are going to be using a method introduced by L.C. Young \cite{young} in 1936.
</p>projecteuclid.org/euclid.rae/1525226424_20180501220024Tue, 01 May 2018 22:00 EDTQuasicontinuous functions with values in Piotrowski spaceshttps://projecteuclid.org/euclid.rae/1525226425<strong>Taras Banakh</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 77--104.</p><p><strong>Abstract:</strong><br/>
A topological space \(X\) is called {\em Piotrowski} if every quasicontinuous map \(f:Z\to X\) from a Baire space \(Z\) to \(X\) has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski spaces to strictly fragmentable, Stegall, and game determined spaces. Also we prove that a Piotrowski Tychonoff space \(X\) contains a dense (completely) metrizable Baire subspace if and only if \(X\) is Baire (Choquet).
</p>projecteuclid.org/euclid.rae/1525226425_20180501220024Tue, 01 May 2018 22:00 EDTOptimal Quantizers for some Absolutely Continuous Probability Measureshttps://projecteuclid.org/euclid.rae/1525226426<strong>Mrinal Kanti Roychowdhury</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 105--136.</p><p><strong>Abstract:</strong><br/>
The representation of a given quantity with less information is often referred to as ‘quantization’ and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit discs, squares, and the real line. For these probability measures the optimal sets of \(n\)-means and the \(n\)th quantization errors are calculated for some positive integers \(n\).
</p>projecteuclid.org/euclid.rae/1525226426_20180501220024Tue, 01 May 2018 22:00 EDTErgodic Properties of Rational Functions that Preserve Lebesgue Measure on ℝhttps://projecteuclid.org/euclid.rae/1525226427<strong>Rachel L. Bayless</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 137--154.</p><p><strong>Abstract:</strong><br/>
We prove that all negative generalized Boole transformations are conservative, exact, pointwise dual ergodic, and quasi-finite with respect to Lebesgue measure on the real line. We then provide a formula for computing the Krengel, Parry, and Poisson entropy of all conservative rational functions that preserve Lebesgue measure on the real line.
</p>projecteuclid.org/euclid.rae/1525226427_20180501220024Tue, 01 May 2018 22:00 EDTDivided Differences, Square Functions, and a Law of the Iterated Logarithmhttps://projecteuclid.org/euclid.rae/1525226428<strong>Artur Nicolau</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 155--186.</p><p><strong>Abstract:</strong><br/>
The main purpose of the paper is to show that differentiability properties of a measurable function defined in the euclidean space can be described using square functions which involve its second symmetric divided differences. Classical results of Marcinkiewicz, Stein and Zygmund describe, up to sets of Lebesgue measure zero, the set of points where a function~\)f\) is differentiable in terms of a certain square function~\)g(f)\). It is natural to ask for the behavior of the divided differences at the complement of this set, that is, on the set of points where \(f\) is not differentiable. In the nineties, Anderson and Pitt proved that the growth of the divided differences of a function in the Zygmund class obeys a version of the classical Kolmogorov's Law of the Iterated Logarithm (LIL). A square function, which is the conical analogue of \(g(f)\) will be used to state and prove a general version of the LIL of Anderson and Pitt as well as to prove analogues of the classical results of Marcinkiewicz, Stein and Zygmund. Sobolev spaces can also be described using this new square function.
</p>projecteuclid.org/euclid.rae/1525226428_20180501220024Tue, 01 May 2018 22:00 EDTMagic Setshttps://projecteuclid.org/euclid.rae/1525226429<strong>Lorenz Halbeisen</strong>, <strong>Marc Lischka</strong>, <strong>Salome Schumacher</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 187--204.</p><p><strong>Abstract:</strong><br/>
\noindent In this paper we study magic sets for certain families \(\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}\) which are subsets \(M\subseteq\mathbb{R}\) such that for all functions \(f,g\in\mathcal{H}\) we have that \(g[M]\subseteq f[M]\Rightarrow f=g\). Specifically we are interested in magic sets for the family \(\mathcal{G}\) of all continuous functions that are not constant on any open subset of \(\mathbb{R}\). We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than \(\mathfrak{c}\) meager sets is still meager, we can also add and remove sets of cardinality less than \(\mathfrak{c}\) without destroying the magic set. \newline \noindent Then we will enlarge the family \(\mathcal{G}\) to a family \(\mathcal{F}\) by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function \(f\:\mathbb{R}\to\mathbb{R}\) is symmetric iff for every \(x\in\mathbb{R}\) we have that \(\lim_{h\downarrow 0}\frac{1}{2}\left(f(x+h)+f(x-h)\right )=f(x)\). For this family of functions we will construct \(2^\mathfrak{c}\) pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than \(\mathfrak{c}\). We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family \(\mathcal{F}\) are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family \(\mathcal{F}\) it is sufficient to assume that the union of less than \(\mathfrak{c}\) meager sets is still meager. So for example Martin's axiom for \(\sigma\)-centered partial orders implies the existence of a magic set.
</p>projecteuclid.org/euclid.rae/1525226429_20180501220024Tue, 01 May 2018 22:00 EDTRandom Cutouts of the Unit Cube with I.U.D Centershttps://projecteuclid.org/euclid.rae/1525226430<strong>Z. Y. Zhu</strong>, <strong>E. M. Dong</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 205--220.</p><p><strong>Abstract:</strong><br/>
Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the d-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0<c<(\beta(d))^{-\frac{1}{d}}\), where \(\beta(d)\) is the volume of the \(d\) dimensional unit ball. We call \([0,1]^d-\bigcup_{n=1}^{\infty} B_n(\omega)\) a random cutout set. In this paper, we present an exposition of Z\)\ddot{a}\)hle cutout model in \cite{Zahle} by a detailed study of such a random cutout set for the purpose of teaching and learning. We show that with probability one Hausdorff dimension of such random cut-out set is at most \(d(1-\beta(d)c^d)\) and frequently equals \(d(1-\beta(d)c^d)\).
</p>projecteuclid.org/euclid.rae/1525226430_20180501220024Tue, 01 May 2018 22:00 EDTOn the Minkowski Sum of Two Curveshttps://projecteuclid.org/euclid.rae/1525226431<strong>Alan Chang</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 221--222.</p><p><strong>Abstract:</strong><br/>
We answer a question posed by Miklós Laczkovich on the Minkowski sum of two curves.
</p>projecteuclid.org/euclid.rae/1525226431_20180501220024Tue, 01 May 2018 22:00 EDTA Note on the Uniqueness Property for Borel G -measureshttps://projecteuclid.org/euclid.rae/1525226432<strong>Alexander Kharazishvili</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 223--234.</p><p><strong>Abstract:</strong><br/>
In terms of a group \(G\) of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a \(G\)-measure on the Borel \(\sigma\)-algebra of this space.
</p>projecteuclid.org/euclid.rae/1525226432_20180501220024Tue, 01 May 2018 22:00 EDTOn the Minkowski Sum of Two Curveshttps://projecteuclid.org/euclid.rae/1525226433<strong>Andrew M. Bruckner</strong>, <strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 235--238.</p><p><strong>Abstract:</strong><br/>
We show that there exists a derivative \(f\colon [0,1]\to[0,1]\) such that the graph of \(f\circ f\) is dense in \([0,1]^2\), so not a \(G_\delta\)-set. In particular, \(f\circ f\) is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative. %neither of Baire class 1 nor a derivative.
</p>projecteuclid.org/euclid.rae/1525226433_20180501220024Tue, 01 May 2018 22:00 EDTA note on the Luzin-Menchoff theoremhttps://projecteuclid.org/euclid.rae/1525226434<strong>Hajrudin Fejzić</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 239--242.</p><p><strong>Abstract:</strong><br/>
A proof of the Luzim-Menchoff theorem.
</p>projecteuclid.org/euclid.rae/1525226434_20180501220024Tue, 01 May 2018 22:00 EDTWhich Integrable Functions Fail to be Absolutely Integrable?https://projecteuclid.org/euclid.rae/1525226435<strong>José Mendoza</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 243--248.</p><p><strong>Abstract:</strong><br/>
An answer to the question of the title is given.
</p>projecteuclid.org/euclid.rae/1525226435_20180501220024Tue, 01 May 2018 22:00 EDTMycielski-Regularity of Gibbs Measures on Cookie-Cutter Setshttps://projecteuclid.org/euclid.rae/1530064959<strong>Jeremiah J. Bass</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 249--262.</p><p><strong>Abstract:</strong><br/>
It has been shown that all Radon probability measures on \(\mathbbm{R}\) are Mycielski-regular, as well as Lebesgue measure on the unit cube and certain self-similar measures. In this paper, these results are extended to Gibbs measures on cookie-cutter sets.
</p>projecteuclid.org/euclid.rae/1530064959_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Choquet Integral in Capacityhttps://projecteuclid.org/euclid.rae/1530064960<strong>Sorin G. Gal</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 263--280.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce and study the new concept of Choquet integral in capacity, which generalizes the Riemann integral in probability and the classical Choquet integral. Properties of this new integral are proved and some applications are presented.
</p>projecteuclid.org/euclid.rae/1530064960_20180626220252Tue, 26 Jun 2018 22:02 EDTMinimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimateshttps://projecteuclid.org/euclid.rae/1530064961<strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 281--292.</p><p><strong>Abstract:</strong><br/>
A rational function of the form \(\frac{x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}}{x_1^{\beta_1} + x_2^{\beta_2}+\cdots + x_n^{\beta_n}}\) is a >Genocchi-Peano example, GPE , provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree \(D(n)\) of a GPE of \(n\)-variables equals \(2\left\lfloor \frac{e^2}{e^2-1} n \right\rfloor+2i\) for some \(i\in\{0,1,2\}\). We also investigate the minimal degree \(D_b(n)\) of a bounded GPE of \(n\)-variables and note that \(D(n)\leq D_b(n)\leq n(n+1)\). Finding better bounds for the numbers \(D_b(n)\) remains an open problem.
</p>projecteuclid.org/euclid.rae/1530064961_20180626220252Tue, 26 Jun 2018 22:02 EDTLipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolationhttps://projecteuclid.org/euclid.rae/1530064962<strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 293--300.</p><p><strong>Abstract:</strong><br/>
We present a simple argument showing that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\), its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the \(C^1\) free interpolation theorem, that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) there exists a continuously differentiable function \(g\colon\mathbb{R}\to\mathbb{R}\) which agrees with \(f\) on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with Lebesgue measure theory.
</p>projecteuclid.org/euclid.rae/1530064962_20180626220252Tue, 26 Jun 2018 22:02 EDTEqui-Riemann and Equi-Riemann Type Integrable Functions with Values in a Banach Spacehttps://projecteuclid.org/euclid.rae/1530064963<strong>Pratikshan Mondal</strong>, <strong>Lakshmi Kanta Dey</strong>, <strong>Sk. Jaker Ali</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 301--324.</p><p><strong>Abstract:</strong><br/>
In this paper we study equi-Riemann and equi-Riemann-type integrability of a collection of functions defined on a closed interval of \(\mathbb{R}\) with values in a Banach space. We obtain some properties of such collections and interrelations among them. Moreover we establish equi-integrability of different types of collections of functions. Finally, we obtain relations among equi-Riemann integrability with other properties of a collection of functions.
</p>projecteuclid.org/euclid.rae/1530064963_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Baire Classification of Strongly Separately Continuous Functions on \(\ell_\infty\)https://projecteuclid.org/euclid.rae/1530064964<strong>Olena Karlova</strong>, <strong>Tomá\v{s} Visnyai</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 325--332.</p><p><strong>Abstract:</strong><br/>
We prove that for any \(\alpha\in[0,\omega_1)\) there exists a strongly separately continuous function \(f:\ell_\infty\rightarrow [0,1]\) such that \(f\) belongs to the \((\alpha+1)\)'th \) /(\alpha+2)\)'th/ Baire class and does not belong to the \(\alpha\)'th Baire class if \(\alpha\) is finite /infinite/.
</p>projecteuclid.org/euclid.rae/1530064964_20180626220252Tue, 26 Jun 2018 22:02 EDTOn the Growth of Real Functions and their Derivativeshttps://projecteuclid.org/euclid.rae/1530064965<strong>J\"urgen Grahl</strong>, <strong>Shahar Nevo</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 333--346.</p><p><strong>Abstract:</strong><br/>
We show that for any \(k\)-times differentiable function \(f:[a,\infty)\to\mathbb{R}\), any integer \(q\ge 0\) and any \(\alpha>1\) the inequality \[ \liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\ldots\cdot \log_q x \cdot |f^{(k)}(x)|}{1+|f(x)|^\alpha}= 0 \] holds and that this result is best possible in the sense that \(\log_q x\) cannot be replaced by \((\log_q x)^\beta\) with any \(\beta>1\).
</p>projecteuclid.org/euclid.rae/1530064965_20180626220252Tue, 26 Jun 2018 22:02 EDTRestricted Families of Projections and Random Subspaceshttps://projecteuclid.org/euclid.rae/1530064966<strong>Changhao Chen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 347--358.</p><p><strong>Abstract:</strong><br/>
We study the restricted families of orthogonal projections in \(\mathbb{R}^{3}\). We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem.
</p>projecteuclid.org/euclid.rae/1530064966_20180626220252Tue, 26 Jun 2018 22:02 EDTSimultaneous Small Coverings by Smooth Functions Under the Covering Property Axiomhttps://projecteuclid.org/euclid.rae/1530064967<strong>Krzysztof C. Ciesielski</strong>, <strong>Juan B. Seoane--Sepúlveda</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 359--386.</p><p><strong>Abstract:</strong><br/>
The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every \(\nu\in\omega\cup\{\infty\}\) there exists a family \(\mathcal{F}_\nu\subset C^\nu(\mathbb{R})\) of cardinality \(\omega_1<\mathfrak{c}\) such that for every \(g\in D^\nu(\mathbb{R})\) the set \(g\setminus \bigcup \mathcal{F}_\nu\) has cardinality \(\leq\omega_1\). Moreover, we show that this result remains true for partial functions \(g\) (i.e., \(g\in D^\nu(X)\) for some \(X\subset\mathbb{R}\)) if, and only if, \(\nu \in\{0,1\}\). The proof of this result is based on the following theorem of independent interest (which, for \(\nu\neq 0\), seems to have been previously unnoticed): for every \(X\subset\mathbb{R}\) with no isolated points, every \(\nu\)-times differentiable function \(g\colon X\to\mathbb{R}\) admits a \(\nu\)-times differentiable extension \(\bar g\colon B\to\mathbb{R}\), where \(B \supset X\) is a Borel subset of \(\mathbb{R}\). The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of \(\mathbb{R}\), which short but fully detailed proof is included. Some open questions are also posed.
</p>projecteuclid.org/euclid.rae/1530064967_20180626220252Tue, 26 Jun 2018 22:02 EDTA Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradienthttps://projecteuclid.org/euclid.rae/1530064968<strong>Anna K. Savvopoulou</strong>, <strong>Christopher M. Wedrychowcz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 387--392.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to give an alternate proof of a result of M. Elekes. We show that if \( f\:\mathbb{R}^2\rightarrow \mathbb{R}\) is a differentiable function with everywhere non-zero gradient, then for every point \(x\in \mathbb{R}^2\) in the level set \(\{x\: \:: f(x)=c\}\) there is a neighborhood \(V\) of \(x\) such that \(\{f=c\}\cap V\) is homeomorphic to an open interval or the union of finitely many open segments passing through a point.
</p>projecteuclid.org/euclid.rae/1530064968_20180626220252Tue, 26 Jun 2018 22:02 EDTS-Limited Shiftshttps://projecteuclid.org/euclid.rae/1530064969<strong>Benjamin Matson</strong>, <strong>Elizabeth Sattler</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 393--416.</p><p><strong>Abstract:</strong><br/>
In this paper, we explore the construction and dynamical properties of \(\mathcal{S}\)-limited shifts. An \(S\)-limited shift is a subshift defined on a finite alphabet \(\mathcal{A} = \{1, \ldots,p\}\) by a set \(\mathcal{S} = \{S_1, \ldots, S_p\}\), where \(S_i \subseteq \mathbb{N}\) describes the allowable lengths of blocks in which the corresponding letter may appear. We give conditions for which an \(\mathcal{S}\)-limited shift is a subshift of finite type or sofic. We give an exact formula for finding the entropy of such a shift and show that an \(\mathcal{S}\)-limited shift and its factors must be intrinsically ergodic. Finally, we give some conditions for which two such shifts can be conjugate, and additional information about conjugate \(\mathcal{S}\)-limited shifts.
</p>projecteuclid.org/euclid.rae/1530064969_20180626220252Tue, 26 Jun 2018 22:02 EDTSome Applications of Order-Embeddings of Countable Ordinals into the Real Linehttps://projecteuclid.org/euclid.rae/1530064970<strong>Leonard Huang</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 417--428.</p><p><strong>Abstract:</strong><br/>
It is a well-known fact that an ordinal \( \alpha \) can be embedded into the real line \( \mathbb{R} \) in an order-preserving manner if and only if \( \alpha \) is countable. However, it would seem that outside of set theory, this fact has not yet found any concrete applications. The goal of this paper is to present some applications. More precisely, we show how two classical results, one in point-set topology and the other in real analysis, can be proven by defining specific order-embeddings of countable ordinals into \( \mathbb{R} \).
</p>projecteuclid.org/euclid.rae/1530064970_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proofhttps://projecteuclid.org/euclid.rae/1530064971<strong>Oswaldo de Oliveira</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 429--444.</p><p><strong>Abstract:</strong><br/>
This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps \(F(x,y)\) defined on a finite-dimensional Euclidean space. There are no hypotheses on the continuity of the partial derivatives of \(F\). The proof employs the mean-value theorem, the intermediate-value theorem, Darboux’s property (the intermediate-value property for derivatives), and determinants theory. The proof avoids compactness arguments, fixed-point theorems, and Lebesgue’s measure. A stronger than the classical version of the Inverse Function Theorem is also shown. Two illustrative examples are given.
</p>projecteuclid.org/euclid.rae/1530064971_20180626220252Tue, 26 Jun 2018 22:02 EDTUniqueness Properties of Harmonic Functionshttps://projecteuclid.org/euclid.rae/1530064972<strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 445--450.</p><p><strong>Abstract:</strong><br/>
We study the zero set of a harmonic function of several real variables. Using the theory of real analytic functions, we analyze such sets. We generalize these results to solutions of elliptic partial differential equations with constant coefficients.
</p>projecteuclid.org/euclid.rae/1530064972_20180626220252Tue, 26 Jun 2018 22:02 EDTAn Earlier Fractal Graphhttps://projecteuclid.org/euclid.rae/1530064973<strong>Harvey Rosen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 451--454.</p><p><strong>Abstract:</strong><br/>
A function \(f:\mathbb{R}\to \mathbb{R}\) is additive if \( f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\). We give examples of an additive function whose graph is fractal.
</p>projecteuclid.org/euclid.rae/1530064973_20180626220252Tue, 26 Jun 2018 22:02 EDTConstructive Analysis on Banach spaceshttps://projecteuclid.org/euclid.rae/1561622429<strong>Tepper L. Gill</strong>, <strong>Timothy Myers</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
Problems requiring analysis in higher-dimensional spaces have appeared naturally in electrical engineering, computer science, mathematics, physics, and statistics. In many cases, these problems focus on objects determined by an infinite number of parameters and/or are defined by functions of an infinite number of variables. They are currently studied using analytic, combinatorial, geometric and probabilistic methods from functional analysis. This paper is devoted to one of the important missing tools, a reasonable (or constructive) theory of Lebesgue measure for separable Banach spaces. A reasonable theory is one that provides: (1) a direct constructive extension of the finite-dimensional theory; and, (2) most (if not all) of the analytic tools available in finite dimensions. We approach this problem by embedding every separable Banach space into \(\mathbb{R}^\infty\) and use the unique \(\sigma {\text{-finite}}\) Lebesgue measure defined on this space as a bridge to the construction of a Lebesgue integral on every separable Banach space as a limit of finite-dimensional integrals. In our first application we define universal versions of Gaussian and Cauchy measure for every separable Banach space, which are absolutely continuous with respect to our Lebesgue measure. As our second application we constructively solve the diffusion equation in infinitely-many variables and introduce the interesting climate model problem of P. D. Thompson defined on infinite-dimensional phase space.
</p>projecteuclid.org/euclid.rae/1561622429_20190627040050Thu, 27 Jun 2019 04:00 EDTStrictly Singular Operators on Banach Latticeshttps://projecteuclid.org/euclid.rae/1561622430<strong>Francisco L. Hernández</strong>, <strong>Evgeny M. Semenov</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 37--48.</p><p><strong>Abstract:</strong><br/>
We survey several properties of the class of strictly singular operators defined on Banach lattices of measurable functions and rearrangement invariant function spaces. The related classes of disjointly strictly singular operators and super strictly singular operators are also discussed. In particular we present interpolation properties of strictly singular operators between \(L_p-L_q\) spaces.
</p>projecteuclid.org/euclid.rae/1561622430_20190627040050Thu, 27 Jun 2019 04:00 EDTDistribution of Polynomials in Many Variables and Nikolskii-Besov Spaceshttps://projecteuclid.org/euclid.rae/1561622431<strong>Vladimir I. Bogachev</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 49--64.</p><p><strong>Abstract:</strong><br/>
We discuss recent results and open problems connected with distributions of polynomials in many or infinitely many variables on spaces with Gaussian measures and Nikolskii--Besov spaces of fractional smoothness related to such distributions.
</p>projecteuclid.org/euclid.rae/1561622431_20190627040050Thu, 27 Jun 2019 04:00 EDTThe Union Problem and The Category Problem of Sets of Uniqueness in the Theory of Orthogonal Serieshttps://projecteuclid.org/euclid.rae/1561622432<strong>Natalia Kholshchevnikova</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 65--76.</p><p><strong>Abstract:</strong><br/>
These problems are considered for a wide range of orthogonal series. That is to say for trigonometric, Walsh, and other series on systems of characters of zero-dimensional compact abelian groups with the second axiom of countability. By this the sets of uniqueness are regarded for one-dimensional and multiple orthogonal series for many types of convergence.
</p>projecteuclid.org/euclid.rae/1561622432_20190627040050Thu, 27 Jun 2019 04:00 EDTDynamics of Certain Distal Actions on Sphereshttps://projecteuclid.org/euclid.rae/1561622433<strong>Riddhi Shah</strong>, <strong>Alok K. Yadav</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 77--88.</p><p><strong>Abstract:</strong><br/>
Consider the action of \(SL(n+1,\mathbb{R})\) on \(\mathbb{S}^n\) arising as the quotient of the linear action on \(\mathbb{R}^{n+1}\setminus\{0\}\). We show that for a semigroup \(\mathfrak{S}\) of \(SL(n+1,\mathbb{R})\), the following are equivalent: \((1)\) \(\mathfrak{S}\) acts distally on the unit sphere \(\mathbb{S}^n\). \((2)\) the closure of \(\mathfrak{S}\) is a compact group. We also show that if \(\mathfrak{S}\) is closed, the above conditions are equivalent to the condition that every cyclic subsemigroup of \(\mathfrak{S}\) acts distally on \(\mathbb{S}^n\). On the unit circle \(\mathbb{S}^1\), we consider the ‘affine’ actions corresponding to maps in \(GL(2,\mathbb{R})\) and discuss the conditions for the existence of fixed points and periodic points, which in turn imply that these maps are not distal.
</p>projecteuclid.org/euclid.rae/1561622433_20190627040050Thu, 27 Jun 2019 04:00 EDTThe Inequality of Milne and its Converse, IIIhttps://projecteuclid.org/euclid.rae/1561622434<strong>Horst Alzer</strong>, <strong>Alexander Kovačec</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 89--100.</p><p><strong>Abstract:</strong><br/>
The discrete version of Milne’s inequality and its converse states that \begin{equation*} (*)\quad \sum_{j=1}^n\frac{w_j}{1-p_j^2} \leq \sum_{j=1}^n\frac{w_j}{1-p_j} \sum_{j=1}^n\frac{w_j}{1+p_j} \leq \Bigl(\sum_{j=1}^n\frac{w_j}{1-p_j^2} \Bigr)^2 \end{equation*} is valid for all \(w_j>0\) \((j=1,...,n)\) with \(w_1+\dots+w_n=1\) and \(p_j\in (-1,1)\) \((j=1,...,n)\). We present new upper and lower bounds for the product \(\sum w/(1-p) \sum w/(1+p)\). In particular, we obtain an improvement of the right-hand side of \((*)\). Moreover, we prove a matrix analogue of our double-inequality.
</p>projecteuclid.org/euclid.rae/1561622434_20190627040050Thu, 27 Jun 2019 04:00 EDTErdős Semi-groups, Arithmetic Progressions, and Szemerédi’s Theoremhttps://projecteuclid.org/euclid.rae/1561622435<strong>Han Yu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 101--118.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce and study a certain type of sub-semigroup of \(\mathbb{R}/\mathbb{Z}\) which turns out to be closely related to Szemerédi’s theorem on arithmetic progressions.
</p>projecteuclid.org/euclid.rae/1561622435_20190627040050Thu, 27 Jun 2019 04:00 EDTHardy-Littlewood Maximal Operator on the Associate Space of a Banach Function Spacehttps://projecteuclid.org/euclid.rae/1561622436<strong>Alexei Yu. Karlovich</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 119--140.</p><p><strong>Abstract:</strong><br/>
Let \(\mathcal{E}(X,d,\mu)\) be a Banach function space over a space of homogeneous type \((X,d,\mu)\). We show that if the Hardy-Littlewood maximal operator \(M\) is bounded on the space \(\mathcal{E}(X,d,\mu)\), then its boundedness on the associate space \(\mathcal{E}'(X,d,\mu)\) is equivalent to a certain condition \(\mathcal{A}_\infty\). This result extends a theorem by Andrei Lerner from the Euclidean setting of \(\mathbb{R}^n\) to the setting of spaces of homogeneous type.
</p>projecteuclid.org/euclid.rae/1561622436_20190627040050Thu, 27 Jun 2019 04:00 EDTSome Characterizations of the Preimage of \(A_{\infty }\) for the Hardy-Littlewood Maximal Operator and Consequenceshttps://projecteuclid.org/euclid.rae/1561622437<strong>Álvaro Corvalán</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 141--166.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to give some characterizations of the weight functions \(w\) such that \(Mw\in A_{\infty }\left( \mathbb{R}^{n}\right) \). We show that, for these \(Mw\) weights, being in \(A_{\infty }\) ensures being in \(A_{1}\). We give a criterion in terms of the local maximal functions \(% m_{\lambda }\) and we present a pair of applications, one of them similar to the Coifman-Rochberg characterization of \(A_{1}\) but using functions of the form \(\left( f^{\#}\right) ^{\delta }\) and \(\left( m_{\lambda }u\right) ^{\delta }\) instead of \(\left( Mf\right) ^{\delta }\).
</p>projecteuclid.org/euclid.rae/1561622437_20190627040050Thu, 27 Jun 2019 04:00 EDTOn the Speed of Convergence in the Strong Density Theoremhttps://projecteuclid.org/euclid.rae/1561622438<strong>Panagiotis Georgopoulos</strong>, <strong>Constantinos Gryllakis</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 167--180.</p><p><strong>Abstract:</strong><br/>
For a compact set \(K\subseteq\mathbb{R}^m\), we have two indexes given under simple parameters of the set \(K\) (these parameters go back to Besicovitch and Taylor in the late 1950’s). In the present paper we prove that with the exception of a single extreme value for each index, we have the following elementary estimate on how fast the ratio in the strong density theorem of Saks will tend to one: \[ \frac{|R\cap K|}{|R|}>1-o\bigg(\frac{1}{|\log d(R)|}\bigg) \qquad \text{for a.e.} \ \ x\in K \ \ \text{and for} \ \ d(R)\to0 \] (provided \(x\in R\), where \(R\) is an interval in \(\mathbb{R}^m\), \(d\) stands for the diameter, and \(|\cdot|\) is the Lebesgue measure).
This work is a natural sequence of [3] and constitutes a contribution to Problem 146 of Ulam [5, p. 245] (see also [8, p.78]) and Erdös' Scottish Book ‘Problems’ [5, Chapter 4, pp. 27-33], since it is known that no general statement can be made on how fast the density will tend to one.
</p>projecteuclid.org/euclid.rae/1561622438_20190627040050Thu, 27 Jun 2019 04:00 EDTThe Weak Integral by Partitions of Unityhttps://projecteuclid.org/euclid.rae/1561622439<strong>Redouane Sayyad</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 181--198.</p><p><strong>Abstract:</strong><br/>
We introduce the notion of the the weak integral by partitions of unity for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and discuss its relation with the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014).
</p>projecteuclid.org/euclid.rae/1561622439_20190627040050Thu, 27 Jun 2019 04:00 EDTOn the Dimension and Measure of Inhomogeneous Attractorshttps://projecteuclid.org/euclid.rae/1561622440<strong>Stuart A. Burrell</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 199--216.</p><p><strong>Abstract:</strong><br/>
A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with notable results by Olsen, Snigireva, Fraser and Käenmäki on inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous self-affine sets. This paper is devoted to filling a significant gap in the dimension theory of inhomogeneous attractors, by studying those formed from arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension of the condensation set and a quantity related to pressure, which we term upper Lipschitz dimension, forms a natural and general upper bound on the dimension. Additionally, we begin a new line of enquiry; the methods developed are used to classify the Hausdorff measure of inhomogeneous attractors. Our results have applications for affine systems with affinity dimension less than or equal to one and systems satisfying bounded distortion, such as conformal systems in dimensions greater than one.
</p>projecteuclid.org/euclid.rae/1561622440_20190627040050Thu, 27 Jun 2019 04:00 EDTOn the Steinhaus Property and Ergodicity via the Measure-Theoretic Density of Setshttps://projecteuclid.org/euclid.rae/1561622441<strong>Alexander Kharazishvili</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 217--228.</p><p><strong>Abstract:</strong><br/>
It is shown how the Steinhaus property and ergodicity of a translation invariant extension \(\mu\) of the Lebesgue measure depend on the measure-theoretic density of \(\mu\)-measurable sets. Some connection of the Steinhaus property with almost convex sets is considered and a translation invariant extension of the Lebesgue measure is presented, for which the generalized Steinhaus property together with the mid-point convexity do not imply the almost convexity.
</p>projecteuclid.org/euclid.rae/1561622441_20190627040050Thu, 27 Jun 2019 04:00 EDTA Didactic Note on Classic Function Spaces and the Fourier Transformhttps://projecteuclid.org/euclid.rae/1561622442<strong>Sandra Lucente</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 1, 229--246.</p><p><strong>Abstract:</strong><br/>
In the present paper we recall the main points of the Fourier Transform developments, in particular the historical origin of the inversion formula. Hence we construct explicit examples of functions in different zones of the range of the Fourier transform in \(L^1\). These can be used as exercises in a basic course of signal processing or harmonic analysis.
</p>projecteuclid.org/euclid.rae/1561622442_20190627040050Thu, 27 Jun 2019 04:00 EDT