Real Analysis Exchange Articles (Project Euclid)
http://projecteuclid.org/euclid.rae
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
http://projecteuclid.org/
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 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 EDTLocal Dimensions of Overlapping Self-Similar Measureshttps://projecteuclid.org/euclid.rae/1588298424<strong>Kathryn E. Hare</strong>, <strong>Kevin G. Hare</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 247--266.</p><p><strong>Abstract:</strong><br/>
We show that any equicontractive, self-similar measure arising from the IFS of contractions \((S_{j})\), with self-similar set \([0,1]\), admits an isolated point in its set of local dimensions provided the images of \(S_{j}(0,1)\) (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include \(m\)-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding \(1/(m+1)\) in the biased case and \(1/m\) in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.
</p>projecteuclid.org/euclid.rae/1588298424_20200430220028Thu, 30 Apr 2020 22:00 EDTA Modification of the Chang-Wilson-Wolff Inequality via the Bellman Functionhttps://projecteuclid.org/euclid.rae/1588298425<strong>Henry D. Riely</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 267--286.</p><p><strong>Abstract:</strong><br/>
We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove \(L^p\) boundedness of the \(\pm 1\) martingale transform. Finally, with Burkholder’s result as a blueprint, we use the Bellman function to prove a new result related to the Chang-Wilson-Wolff Inequality.
</p>projecteuclid.org/euclid.rae/1588298425_20200430220028Thu, 30 Apr 2020 22:00 EDTRiemann Summability of Trigonometric Series and Riemann Derivatives of Real Functionshttps://projecteuclid.org/euclid.rae/1588298426<strong>S. N. Mukhopadhyay</strong>, <strong>S. Ray</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 287--304.</p><p><strong>Abstract:</strong><br/>
A relation between Riemann summability and Riemann derivative is established and necessary and sufficient conditions for Riemann summability of trigonometric series are obtained
</p>projecteuclid.org/euclid.rae/1588298426_20200430220028Thu, 30 Apr 2020 22:00 EDTFractional Hermite-Hadamard Type Integral Inequalities for Functions whose Modulus of the Mixed Derivatives are Co-ordinated Extended \(\left( s_{1},m_{1}\right) \(- \(\left( s_{2},m_{2}\right) \(-Preinvexhttps://projecteuclid.org/euclid.rae/1588298427<strong>Wahida Kaidouchi</strong>, <strong>Badreddine Meftah</strong>, <strong>Meryem Benssaad</strong>, <strong>Sarra Ghomrani</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 305--332.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish a new fractional identity involving a functions of two independent variables, and then we derive some fractional Hermite-Hadamard type integral inequalities for functions whose modulus of the mixed derivatives are co-ordinated extended \(\left( s_{1},m_{1}\right) \)-\(\left( s_{2},m_{2}\right) \)-preinvex.
</p>projecteuclid.org/euclid.rae/1588298427_20200430220028Thu, 30 Apr 2020 22:00 EDTHahn-Banach-type Theorems and Applications to Optimization for Partially Ordered Vector Space-Valued Invariant Operatorshttps://projecteuclid.org/euclid.rae/1588298428<strong>Antonio Boccuto</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 333--368.</p><p><strong>Abstract:</strong><br/>
We prove sandwich, Hahn-Banach, Fenchel duality theorems and a version of the Moreau-Rockafellar formula for invariant partially ordered vector space-valued operators. As consequences and applications, we give some versions of Farkas and Kuhn-Tucker-type optimization results and separation theorems, we prove the equivalence of these results and give a further application to Tarski-type theorems and probability measures defined on suitable product spaces.
</p>projecteuclid.org/euclid.rae/1588298428_20200430220028Thu, 30 Apr 2020 22:00 EDTApproximations by Differences of Lower Semicontinuous and Finely Continuous Functionshttps://projecteuclid.org/euclid.rae/1588298429<strong>Jaroslav Lukeš</strong>, <strong>Petr Pošta</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 369--382.</p><p><strong>Abstract:</strong><br/>
A classical theorem of W.Sierpiński, S. Mazurkiewicz and S.Kempisty says that the class of all differences of lower semicontinuous functions is uniformly dense in the space of all Baire-one functions. We show a generalization of this result to the case when finely continuous functions of either density topologies or both linear and nonlinear potential theory are involved. Moreover, we examine which topological properties play a crucial role when deriving approximation theorems in more general situations.
</p>projecteuclid.org/euclid.rae/1588298429_20200430220028Thu, 30 Apr 2020 22:00 EDTFourier Method Revised to Solve Partial Differential Equations and Prove Uniqueness at One Strokehttps://projecteuclid.org/euclid.rae/1588298430<strong>Rodrigo López Pouso</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 383--390.</p><p><strong>Abstract:</strong><br/>
We present a novel application of Fourier analysis for solving PDEs which is much faster than the usual separation of variables method and, moreover, it implies uniqueness of the obtained solution at the same time.
</p>projecteuclid.org/euclid.rae/1588298430_20200430220028Thu, 30 Apr 2020 22:00 EDTThe Radon Nikodym Property and Multipliers of \(\mathcal{HK}\)-Integrable Functionshttps://projecteuclid.org/euclid.rae/1588298431<strong>Savita Bhatnagar</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 391--402.</p><p><strong>Abstract:</strong><br/>
We study the space of vector valued multipliers of strongly Henstock-Kurzweil \((\mathcal{SHK})\) integrable functions. We prove that if \(X\) is a commutative Banach algebra, with identity \(e\) of norm one, satisfying Radon-Nikodym property and \(g:[a,b] \rightarrow X\) is of strong bounded variation, then the multiplication operator defined by \(M_g(f)=fg\) maps \(\mathcal{SHK}\) to \(\mathcal{SHK}.\) We also investigate the problems when the domain is \(\mathcal{HK}\) or when \(X\) satisfies weak Radon-Nikodym property.
</p>projecteuclid.org/euclid.rae/1588298431_20200430220028Thu, 30 Apr 2020 22:00 EDTVerifying Differentiability Without Calculating the Derivativehttps://projecteuclid.org/euclid.rae/1588298432<strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 403--426.</p><p><strong>Abstract:</strong><br/>
We study various real-variable techniques for determining whether a function is differentiable without actually calculating a derivative. These include: approximation theory \item Fourier analysis Sobolev spaces \item Poisson integral finite differences Campanato-Morrey theory Landau's inequalities In most cases complete proofs are given.
</p>projecteuclid.org/euclid.rae/1588298432_20200430220028Thu, 30 Apr 2020 22:00 EDTA Descriptive Definition of the Backwards Itô-Henstock Integralhttps://projecteuclid.org/euclid.rae/1588298433<strong>Ricky F. Rulete</strong>, <strong>Mhelmar A. Labendia</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 427--444.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduced the backwards derivative of a Hilbert space-valued function and formulate a version of Fundamental Theorem for the backwards Itô-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process.
</p>projecteuclid.org/euclid.rae/1588298433_20200430220028Thu, 30 Apr 2020 22:00 EDTA Bridge Between Unit Square and Single Integrals for Real Functions of the Form \(\,f(x \cdot y)\)https://projecteuclid.org/euclid.rae/1588298434<strong>Fábio M. S. Lima</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 44, Number 2, 445--462.</p><p><strong>Abstract:</strong><br/>
Sondow and co-workers have employed a key change of variables in order to evaluate double integrals over the unit square \([0,1] \times [0,1]\) in exact closed-form. Motivated by their results, I introduce here a change of variables which creates a ‘bridge’ between integrals of the form \(\,\int_0^1\!\!\int_0^1{f(x \cdot y)~dx \, dy}\,\) and single integrals of the form \(\int_0^1{f(p)\,\ln{p}~d p}\). This allows for prompt closed-form evaluations of several interesting integrals, including some of those investigated recently by Sampedro. I also show that the bridge holds when the intervals of integration are changed from \([0,1]\) to \([1,\infty)\). Finally, a generalization for higher dimensions is proved, which reveals an interesting link of those integrals to Mellin’s transform.
</p>projecteuclid.org/euclid.rae/1588298434_20200430220028Thu, 30 Apr 2020 22:00 EDTCommutators, BMO, Hardy Spaces and Factorization: A Surveyhttps://projecteuclid.org/euclid.rae/1588989619<strong>Brett D. Wick</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 1--28.</p><p><strong>Abstract:</strong><br/>
In this survey we discuss the connection between commutator operators and functions of bounded mean oscillation, and at the same time outline the parallel story of the real Hardy space and weak factorization of these functions. We provide motivation as to why these questions are interesting and highlight the many different methods of proof that exist for proving results of the type discussed.
</p>projecteuclid.org/euclid.rae/1588989619_20200508220024Fri, 08 May 2020 22:00 EDTHausdorff Dimensions for Graph-directed Measures Driven by Infinite Rooted Treeshttps://projecteuclid.org/euclid.rae/1588989620<strong>Kazuki Okamura</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 29--72.</p><p><strong>Abstract:</strong><br/>
We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite \(N\)-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierpínski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham’s functional equations driven by linear fractional transformations.
</p>projecteuclid.org/euclid.rae/1588989620_20200508220024Fri, 08 May 2020 22:00 EDTA Simple Closed Curve in \(\mathbb{R}^3\) Whose Convex Hull Equals the Half-sum of the Curve with Itselfhttps://projecteuclid.org/euclid.rae/1588989621<strong>Mikhail Patrakeev</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 73--84.</p><p><strong>Abstract:</strong><br/>
If \(\Gamma\) is the range of a Jordan curve that bounds a convex set in the plane, then \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma),\) where \(+\) is the Minkowski sum and \(\mathsf{co}\) is the convex hull. Answering a question of V. N. Ushakov, we construct a simple closed curve in \(\mathbb{R}^3\) whose range \(\Gamma\) satisfies \(\frac{1}{2}(\Gamma+\Gamma)=\mathsf{co}(\Gamma)=[0,1]^3.\) Also we show that such a simple closed curve cannot be rectifiable.
</p>projecteuclid.org/euclid.rae/1588989621_20200508220024Fri, 08 May 2020 22:00 EDTAccessible Values for the Assouad and Lower Dimensions of Subsetshttps://projecteuclid.org/euclid.rae/1588989622<strong>Changhao Chen</strong>, <strong>Meng Wu</strong>, <strong>Wen Wu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 85--100.</p><p><strong>Abstract:</strong><br/>
Let \(E\) be a subset of a doubling metric space \((X,d)\). We prove that for any \(s\in [0, \dim_{A}E]\), where \(\dim_{A}\) denotes the Assouad dimension, there exists a subset \(F\) of \(E\) such that \(\dim_{A}F=s\). We also show that the same statement holds for the lower dimension \(\dim_L\).
</p>projecteuclid.org/euclid.rae/1588989622_20200508220024Fri, 08 May 2020 22:00 EDTDouble Lusin Condition and Convergence Theorems for the Backwards Itô-Henstock Integralhttps://projecteuclid.org/euclid.rae/1588989623<strong>Ricky F. Rulete</strong>, <strong>Mhelmar A. Labendia</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 101--126.</p><p><strong>Abstract:</strong><br/>
In this paper, we formulate an equivalent definition of the backwards Itô-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued \(Q\)-Wiener process using double Lusin condition. Moreover, we establish some versions of convergence theorems for this integral.
</p>projecteuclid.org/euclid.rae/1588989623_20200508220024Fri, 08 May 2020 22:00 EDTCauchy’s Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimalshttps://projecteuclid.org/euclid.rae/1588989624<strong>Jacques Bair</strong>, <strong>Piotr Błaszczyk</strong>, <strong>Peter Heinig</strong>, <strong>Vladimir Kanovei</strong>, <strong>Mikhail G. Katz</strong>, <strong>Thomas McGaffey</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 127--150.</p><p><strong>Abstract:</strong><br/>
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy’s infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy’s work challenges received views on Cauchy’s role in the history of analysis and geometry. We demonstrate the viability of Cauchy’s infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence.
</p>projecteuclid.org/euclid.rae/1588989624_20200508220024Fri, 08 May 2020 22:00 EDTChange of Variable Formulas for Riemann Integralshttps://projecteuclid.org/euclid.rae/1588989625<strong>Alberto Torchinsky</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 151--172.</p><p><strong>Abstract:</strong><br/>
We consider general formulations of the change of variable formula for Riemann integrals.
</p>projecteuclid.org/euclid.rae/1588989625_20200508220024Fri, 08 May 2020 22:00 EDTTypes of Convergence Which Preserve Continuityhttps://projecteuclid.org/euclid.rae/1588989626<strong>Simon Reinwand</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 173--204.</p><p><strong>Abstract:</strong><br/>
In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space \(BV\) of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space \(BV\) into itself are automatically continuous.
</p>projecteuclid.org/euclid.rae/1588989626_20200508220024Fri, 08 May 2020 22:00 EDTThe Spectral Theorem from Scratchhttps://projecteuclid.org/euclid.rae/1588989627<strong>Gabriel Nagy</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 1, 205--234.</p><p><strong>Abstract:</strong><br/>
We provide an elementary approach to the development of the continuous functional calculus both for a single bounded normal operator, as well as for commuting tuples of bounded self-adjoint operators on a Hilbert space.
</p>projecteuclid.org/euclid.rae/1588989627_20200508220024Fri, 08 May 2020 22:00 EDTRecent developments in two weight testing theoryhttps://projecteuclid.org/euclid.rae/1593482441<strong>Eric T. Sawyer</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 235--264.</p><p><strong>Abstract:</strong><br/>
This paper is a survey of recent developments in two weight testing theory arising in joint projects with Tuomas Hyötnen, Kangwei Li, Chun-Yen Shen, Ignacio Uriarte-Tuero, Robert Rahm, and Brett Wick. An hour lecture on this work was presented in a talk at the Alamo Symposium in San Antonio, Texas in June of 2019. Our focus is the half-century-old research area of two weight inequalities for Calderón-Zygmund and other classical operators \(T\).
</p>projecteuclid.org/euclid.rae/1593482441_20200629220050Mon, 29 Jun 2020 22:00 EDTBest Local Weighted Approximation. An Approach with Abstract Seminormshttps://projecteuclid.org/euclid.rae/1593482442<strong>Claudia V. Ridolfi</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 265--282.</p><p><strong>Abstract:</strong><br/>
We extend results given by Zó and Cuenya in 2007 about a general approach to problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials of some degree. This approach is called best local approximation. We consider a weighted local approximation of a vector-valued function on the origin and a weighted best local approximation of a real-valued function on several points, similar to classical problems in best local approximation with balanced neighborhood.
</p>projecteuclid.org/euclid.rae/1593482442_20200629220050Mon, 29 Jun 2020 22:00 EDTThe Continuous Primitive Integral in the Planehttps://projecteuclid.org/euclid.rae/1593482443<strong>Erik Talvila</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 283--326.</p><p><strong>Abstract:</strong><br/>
An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions \(F(x,y)\) defined on the extended real plane \([-\infty,\infty]^2\) that vanish when \(x\) or \(y\) is \(-\infty\). With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative \(\partial^2/(\partial x\partial y)\) of this space of primitives. If \(f=\partial^2/(\partial x\partial y) F\) then the integral over interval \([a,b]\times [c,d] \subseteq[-\infty,\infty]^2\) is \(\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)\) and \(\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)\). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is \({\lVert f\rVert}={\lVert F\rVert}_\infty\) where \(F\) is the unique primitive of \(f\). The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both \(L^1\) and the space of Henstock--Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in \({\lVert \cdot\rVert}_\infty\) are also inherited by the integrable distributions. It is shown that the dual space is the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder’s inequality, second mean value theorem, Fubini’s theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in \({\mathbb R}^n\) are sketched out.
</p>projecteuclid.org/euclid.rae/1593482443_20200629220050Mon, 29 Jun 2020 22:00 EDTNon-Differentiability of the Convolution of Differentiable Real Functionshttps://projecteuclid.org/euclid.rae/1593482444<strong>Krzysztof C. Ciesielski</strong>, <strong>Pablo Jiménez-Rodríguez</strong>, <strong>Gustavo A. Mu\~{n}oz-Fernández</strong>, <strong>Juan B. Seoane-Sepúlveda</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 327--338.</p><p><strong>Abstract:</strong><br/>
We provide an example of two \(2\)-periodic everywhere differentiable functions \(f,g\colon\mathbb{R}\to\mathbb{R}\) whose convolution \(f*g\) fails to be differentiable at every point of some perfect (thus, uncountable) set \(P\subset\mathbb{R}\). This shows that the convolution operator can actually destroy the differentiability of these maps, rather than introducing additional smoothness (as it is usually the case). New directions and open problems are also posed.
</p>projecteuclid.org/euclid.rae/1593482444_20200629220050Mon, 29 Jun 2020 22:00 EDTBoundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaceshttps://projecteuclid.org/euclid.rae/1593482445<strong>Rza Mustafayev</strong>, <strong>Nevin Bilgiçli</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 339--374.</p><p><strong>Abstract:</strong><br/>
The boundedness of the weighted iterated Hardy-type operators \(T_{u,b}\) and \(T_{u,b}^*\) involving suprema from weighted Lebesgue space \(L_p(v)\) into weighted Cesàro function spaces \({\operatorname{Ces}}_{q}(w,a)\) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator \(R_u\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator \(P_{u,b }\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. Under additional condition on \(u\) and \(b\), we are able to characterize the boundedness of weighted iterated Hardy-type operator \(T_{u,b}\) involving suprema from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function \(M_{\gamma}\) from \(\Lambda^p(v)\) into \(\Gamma^q(w)\).
</p>projecteuclid.org/euclid.rae/1593482445_20200629220050Mon, 29 Jun 2020 22:00 EDTThe Dynamics of a Typical Measurable Function are Determined on a Zero Measure Sethttps://projecteuclid.org/euclid.rae/1593482446<strong>T. H. Steele</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 375--386.</p><p><strong>Abstract:</strong><br/>
Let \(A\) be a zero measure dense \(G_{\delta }\) subset of \(I=[0,1]\), with \(% \mathcal{M}\) the set of measurable self-maps of \(I\). There exists a residual set \(\mathcal{R}\subset \mathcal{M}\) such that for each \(f\) in \(\mathcal{R}\), the range of \(f\) is contained in \(A\), and the function \(f\) is one-to-one. Moreover, there exists \(h:I\rightarrow I\), a Baire-2 function, such that \(f(x)=h(x)\) \(a.e.\), and for any \(x\in I\), the trajectory \(\tau (x,h)\) is \(\infty \)-adic, so that the \(\omega \)-limit set \(\omega (x,h)\) is a Cantor set. Since the range of \(f\) is contained in \(A\), it follows that for any \(x\) in \(I\), there exists \(y\) in \(A\) such that the trajectory \(\tau (f(x),f)=\tau (y,f)\subset A\). Speaking loosely, the dynamical structures of \(f\) are completely determined by its behavior on the set \(A\).
</p>projecteuclid.org/euclid.rae/1593482446_20200629220050Mon, 29 Jun 2020 22:00 EDTOn the Density of the Thinnest Covering of \(\mathbb{R}^n\)https://projecteuclid.org/euclid.rae/1593482447<strong>Soon-Mo Jung</strong>, <strong>Doyun Nam</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 387--400.</p><p><strong>Abstract:</strong><br/>
Let \(K\) be a compact convex subset of \(\mathbb{R}^n\) whose Lebesgue measure is positive and which includes the origin of the coordinate system. In this paper, the density of the thinnest covering of \(\mathbb{R}^n\) by translates of \(K\) will be transformed into another form that seems easier to use. Finally, in the discussion section of this paper, we will look at the applications of the main results.
</p>projecteuclid.org/euclid.rae/1593482447_20200629220050Mon, 29 Jun 2020 22:00 EDTStability of \(n\)-th Order Flett’s and Sahoo-Riedel’s Pointshttps://projecteuclid.org/euclid.rae/1593482448<strong>Dariusz Sokolowski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 401--410.</p><p><strong>Abstract:</strong><br/>
We show the Hyers-Ulam stability for points in the extended versions of Flett and Sahoo-Riedel theorems.
</p>projecteuclid.org/euclid.rae/1593482448_20200629220050Mon, 29 Jun 2020 22:00 EDTCharacterizing the Coordinate Functions of Space Filling Curveshttps://projecteuclid.org/euclid.rae/1593482449<strong>James Foran</strong>, <strong>Judit Kardos</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 411--424.</p><p><strong>Abstract:</strong><br/>
The coordinate functions, \(f\) and \(g\), of a space filling curve are continuous functions from \([0,1]\) to \([0,1]\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto the unit square. In the paper Coordinate Functions of Space Filling Curves written by Foran, several necessary conditions for a continuous function \(f\) are given for there to be a continuous \(g\) so that \(F(t)=(f(t),g(t))\) maps \([0,1]\) onto \([0,1]^2\). In this paper, we define a new condition for \(f\) that is both necessary and sufficient to assure that \(f\) has a matching coordinate function \(g\) such that \(F(t)=(f(t),g(t))\) fills the square.
</p>projecteuclid.org/euclid.rae/1593482449_20200629220050Mon, 29 Jun 2020 22:00 EDTAsymptotic Constants in Averaged Hölder Inequalitieshttps://projecteuclid.org/euclid.rae/1593482450<strong>Gerd Herzog</strong>, <strong>Peer Chr. Kunstmann</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 425--438.</p><p><strong>Abstract:</strong><br/>
For \(0 \lt q \le p\) we take \(L^\beta\)-averages over the Hölder inequality between the \(l^q\)-norm and the \(l^p\)-norm in \(\mathbb{R}^n\). We obtain precise limits as \(n\to\infty\) for the \(l^p\)-unit ball and, in case \(p\ge1\), also for the \(l^p\)-unit sphere, which coincide and are independent of \(\beta>0\). These are consequences of more general results on the asymptotic behavior of corresponding integrals over balls and spheres of certain bounded measurable functions.
</p>projecteuclid.org/euclid.rae/1593482450_20200629220050Mon, 29 Jun 2020 22:00 EDTGeneralized almost statistical convergencehttps://projecteuclid.org/euclid.rae/1593482451<strong>Absos Ali Shaikh</strong>, <strong>Biswa Ranjan Datta</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 439--452.</p><p><strong>Abstract:</strong><br/>
The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.
</p>projecteuclid.org/euclid.rae/1593482451_20200629220050Mon, 29 Jun 2020 22:00 EDTThe Intersection of Typical Besicovitch Sets with Lineshttps://projecteuclid.org/euclid.rae/1593482452<strong>Tamás Kátay</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 453--464.</p><p><strong>Abstract:</strong><br/>
We show that a typical Besicovitch set \(B\) has intersections of measure zero with every line not contained in it. Moreover, every line in \(B\) intersects the union of all the other lines in \(B\) in a set of measure zero.
</p>projecteuclid.org/euclid.rae/1593482452_20200629220050Mon, 29 Jun 2020 22:00 EDTA Note on the Existence of Real Two-dimensional Symmetric Subspaces of \(L^p[-1,1]\)https://projecteuclid.org/euclid.rae/1593482453<strong>Grzegorz Lewicki</strong>, <strong>Michael Prophet</strong>, <strong>William E. Wood</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 465--480.</p><p><strong>Abstract:</strong><br/>
An \(n\)-dimensional normed space is said to be symmetric if there exists a basis \(v_1, \dots, v_n\) such that \[ \left\| \sum_{i} |c_i| v_i \right\| = \left\| \sum_i c_{\sigma(i) } v_i \right\| \] for all \(c_1, \dots, c_n \in \mathbb R\) and for every \(\sigma \in \text{Perm}(n)\), the set of all permutations on \(n\) elements. In this paper we consider the existence of 2-dimensional symmetric subspaces of \(L^p[-1,1]\), spanned by \(t^k_e\) and \(t^k_o\) for even and odd integers \(k_e\) and \(k_o\). As the question of symmetry is basis-specific, we organize our consideration by (so-called) diagonal, triangular and symmetric bases. We find that, outside the \(p=4\) and \(p=2\) cases, symmetry is not possible for the bases considered.
</p>projecteuclid.org/euclid.rae/1593482453_20200629220050Mon, 29 Jun 2020 22:00 EDTA Generalization of the Riemann-Lebesgue Theorem for Riemann Integrabilityhttps://projecteuclid.org/euclid.rae/1593482454<strong>Otgonbayar Uuye</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 481--486.</p><p><strong>Abstract:</strong><br/>
A classical theorem of Riemann and Lebesgue says that a bounded function defined on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. In this note, we generalize their result and show that the difference between the upper and lower Riemann integrals of a not necessarily Riemann integrable function equals the upper Riemann integral of its oscillation function.
</p>projecteuclid.org/euclid.rae/1593482454_20200629220050Mon, 29 Jun 2020 22:00 EDTCorrigendum to: On the Speed of Convergence in the Strong Density Theoremhttps://projecteuclid.org/euclid.rae/1593482455<strong>Panagiotis Georgopoulos</strong>, <strong>Constantinos Gryllakis</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 45, Number 2, 487--488.</p><p><strong>Abstract:</strong><br/>
This note corrects two misleading typographical errors in On the speed of convergence in the strong density theorem by P. Georgopoulos and C. Cryllakis, Real Anal. Exch. 44(1) 2019.
</p>projecteuclid.org/euclid.rae/1593482455_20200629220050Mon, 29 Jun 2020 22:00 EDT