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## The approximate variation to pointwise selection principles

Cornell University Library, NY, USA
,
2019.
No. arXiv: 1910.08490.

Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation $V_\varepsilon(f)$ was introduced by Fra{\v n}kov{\'a} [Math. Bohem. 116 (1991), 20--59] for intervals $T=[a,b]$ in $\mathbb{R}$ and $M=\mathbb{R}^N$ and extended to the general case by Chistyakov and Chistyakova [Studia Math. 238 (2017), 37--57]. We prove directly the following basic pointwise selection principle: If a sequence of functions $\{f_j\}_{j=1}^\infty$ from $M^T$ is such that the closure in $M$ of the set $\{f_j(t):j\in\mathbb{N}\}$ is compact for all $t\in T$ and $\limsup_{j\to\infty}V_\varepsilon(f_j)$ is finite for all $\varepsilon>0$, then it contains a subsequence, which converges pointwise on $T$ to a bounded regulated function $f\in M^T$. We establish several variants of this result for sequences of regulated and nonregulated functions, for functions with values in reflexive separable Banach spaces, for the almost everywhere convergence and weak pointwise convergence of extracted subsequences, and comment on the necessity of assumptions in the selection principles. The sharpness of all assertions is illustrated by examples.

Keywords: metric spaceметрическое пространствоselection principlepointwise convergenceпринцип выборапоточечная сходимостьregulated functionрегулярная функцияWeak convergenceHelly's theoremслабая сходимостьapproximate variationтеорема Хеллиаппроксимативная вариация

Publication based on the results of:

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, / Cornell University, NY, USA. Series arXiv [math.FA] "Functional Analysis". 2020. No. 2010.11410.

Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=\,the family of all functions from $T$ into $X$), the {\em approximate variation\/} of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ ...

Added: October 23, 2020

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, Studia Mathematica 2017 Vol. 238 No. 1 P. 37-57

Given a subset $T$ of the reals $R$ and a metric space $M$, we introduce a nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^T$ (the set of all functions from $T$ into $M$), called the joint modulus of variation. We prove that if two sequences $\{f_j\}$ and $\{g_j\}$ of functions from $M^T$ are such that $\{f_j\}$ ...

Added: May 11, 2017

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, / Cornell University. Series math "arxiv.org". 2016. No. 1601.07298.

Given a subset T of real numbers and a metric space M, we introduce a nondecreasing sequence {v_n} of pseudometrics on the set M^T of all functions from T into M, called the joint modulus of variation. We prove that if two sequences of functions {f_j} and {g_j} from M^T are such that {f_j} is ...

Added: February 12, 2016

V. V. Chistyakov, S. A. Chistyakova, Lobachevskii Journal of Mathematics 2022 Vol. 43 No. 3 P. 550-563

Given a Hausdorff uniform space X with the countable gage of pseudometrics of the
uniformity of X, we introduce a concept of the approximate variation of a function f mapping a
subset T of the reals intoX: this is the infimum of the family of Jordan-type variations of all functions
g : T → X which differ from ...

Added: April 30, 2022

С.А.Чистякова, В.В.Чистяков, В кн. : Труды Математического центра имени Н.И.Лобачевского. Т. 54: Теория функций, ее приложения и смежные вопросы.: Каз. : Издательство Казанского математического общества и Академии наук РТ, 2017. С. 399-402.

Given a closed interval $I=[a,b]$ and a metric space $(M,d)$, we introduce a
nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^I$ (the set of all
functions from $I$ into $M$), called the {\it joint modulus of variation}. We show that
if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^I$ are such that
$\{f_j\}$ is pointwise relatively compact on $I$, ...

Added: August 29, 2017

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, Journal of Mathematical Analysis and Applications 2017 Vol. 452 No. 2 P. 970-989

We introduce a pseudometric TV on the set M^X of all functions mapping a rectangle X on the plane R^2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and {gj} of functions from M^X are such that {fj} is pointwise precompact on X, {gj} is pointwise ...

Added: April 13, 2017

Vyacheslav V. Chistyakov, Springer, 2021

This book addresses the minimization of special lower semicontinuous functionals
over (closed) balls in metric spaces, called the approximate variation. The new
notion of approximate variation contains more information about the bounded
variation functional and has the following features: the infimum in the definition
of approximate variation is not attained in general and the total Jordan variation
of a function ...

Added: October 29, 2021

Vyacheslav V. Chistyakov, Tretyachenko Y. V., Journal of Mathematical Analysis and Applications 2013 Vol. 402 No. 2 P. 648-659

Given a rectangle in the real Euclidean n-dimensional space and two maps f and g defined on it and taking values in a metric semigroup, we introduce the notion of the total joint variation TV(f , g) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) ...

Added: August 29, 2013

Chistyakov V., Tretyachenko Y., Journal of Mathematical Analysis and Applications 2010 Vol. 370 No. 2 P. 672-686

Given two points a=(a1,…,an) and b=(b1,…,bn) from Rn with a<b componentwise and a map f from the rectangle <img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si5.gif" /> into a metric semigroup M=(M,d,+), we study properties of the total variation<img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si7.gif" /> of f on <img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si8.gif" /> introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of ...

Added: November 22, 2012

Tretyachenko Y., Известия высших учебных заведений. Математика 2010 № 5 С. 41-54

In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent ...

Added: September 28, 2012

Tretyachenko Y., Russian Mathematics 2010 Vol. 54 No. 5 P. 35-46

In this paper we consider sequences of functions that are defined on a subset of the real line and take on values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of ...

Added: November 14, 2012

Chistyakov V., Tretyachenko Y., Journal of Mathematical Analysis and Applications 2010 Vol. 369 No. 1 P. 82-93

Given a=(a1,…,an), b=(b1,…,bn)∈Rn with ab componentwise and a map f from the rectangle Iab=[a1,b1]×⋯×[an,bn] into a metric semigroup M=(M,d,+), denote by TV(f,Iab) the Hildebrandt–Leonov total variation of f on Iab, which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted ...

Added: January 9, 2013

Klimenko A. V., Bufetov A., Series C., / Cornell University. Series math "arxiv.org". 2018. No. 1805.11743.

Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving ...

Added: September 18, 2018

Vyacheslav V. Chistyakov, / Cornell University, NY, USA. Series arXiv [math.FA] "Functional Analysis". 2019. No. 1901.09722.

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions ...

Added: January 29, 2019

Bogachev V., Malofeev I., Journal of Mathematical Analysis and Applications 2020 Vol. 486 No. 1 (123883) P. 1-30

We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always ...

Added: October 13, 2020

Klimenko A. V., Bufetov A. I., Series C., Commentarii Mathematici Helvetici 2023

We prove pointwise convergence of spherical averages for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen--Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving ...

Added: October 31, 2022

Delon J., Salomon J., Sobolevski A., Journal of Mathematical Sciences 2012 Vol. 181 No. 6 P. 782-791

We consider a minimum-weight perfect matching problem on the line and establish a "bottom-up" recursion relation for partial minimum-weight matchings. ...

Added: May 11, 2012

Jerónimo-Castro J., Magazinov A., Soberón P., Discrete Mathematics 2015 Vol. 338 No. 9 P. 1577-1585

In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol’nikov posed the following problem. Consider three non-empty families of translates of a convex compact set K in the plane. Suppose that every two translates from different families have a point of intersection. Is it always true that one of the families can be pierced by a ...

Added: October 4, 2018

Chistyakov V., Nonlinear Analysis 2010 Vol. 72 No. 1 P. 15-30

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x0∈X, the set Xw={x∈X:limλ→∞w(λ,x,x0)=0} is a ...

Added: January 25, 2013

Vyacheslav V. Chistyakov, Switzerland : Springer, 2015

Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as generalized velocity fields and serve two important purposes: generate metric spaces in a unified manner and provide a ...

Added: December 31, 2015

Igor L. Kheifets, The Econometrics Journal, by the Royal Economic Society 2015 Vol. 18 No. 1 P. 67-94

We propose a new adequacy test and a graphical evaluation tool for nonlinear dynamic models. The proposed techniques can be applied in any set‐up where parametric conditional distribution of the data is specified and, in particular, to models involving conditional volatility, conditional higher moments, conditional quantiles, asymmetry, Value at Risk models, duration models, diffusion models, ...

Added: February 23, 2021

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017