Abstract
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 into \(X\): this is the infimum of the family of Jordan-type variations of all functions \(g:T\to X\) which differ from \(f\) in each uniform pseudometric, generated by a pseudometric from the gage, not greater than \(\varepsilon>0\). We prove the following compactness theorem in the topology of pointwise convergence: if a pointwise relatively sequentially compact sequence of functions is such that the limit superior of its approximate variations is finite for all pseudometrics in the gage and all \(\varepsilon>0\), then it contains a subsequence which converges pointwise on the domain \(T\) to a bounded regulated function (in a generalized sense). We illustrate this result by appropriate sharp examples and present a new characterization of uniform space valued regulated functions in terms of the approximate variation.
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Notes
We recall Helly’s selection principle for monotone functions in Section 4.
In Section 2, we adopt a more general definition of regulated functions.
As usual, \(V_{\varepsilon+0,\,p}(f,T)\) and \(V_{\varepsilon-0,\,p}(f,T)\) are the limits from the right and from the left of the function \(\varepsilon\mapsto V_{\varepsilon,\,p}(f,T)\), respectively.
Recall that \(g\in X^{I}\) is a step function if, for some partition \(a=t_{0}<t_{1}<t_{2}<\ldots<t_{m-1}<t_{m}=b\) of \(I=[a,b]\), \(g\) takes a constant value on each open interval \((t_{i-1},t_{i})\), \(i=1,2,\ldots,m\). Clearly, \(g\in\text{BV}_{p}(I;X)\) for all \(p\in\mathcal{P}\).
REFERENCES
A. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures (Atlantis, World Scientific, Amsterdam, 2008).
G. Aumann, Reelle Funktionen (Springer, Berlin, 1954).
V. Barbu and Th. Precupanu, Convexity and Optimization in Banach Spaces, 2nd ed. (Reidel, Dordrecht, 1986).
N. Bourbaki, General Topology (Springer, Berlin, 1987), Chaps. 1–4.
Z. A. Chanturiya, ‘‘The modulus of variation of a function and its application in the theory of Fourier series,’’ Sov. Math. Dokl. 15, 67–71 (1974).
V. V. Chistyakov, ‘‘On the theory of multivalued mappings of bounded variation of one real variable,’’ Sb. Math. 189, 797–819 (1998).
V. V. Chistyakov, ‘‘Selections of bounded variation,’’ J. Appl. Anal. 10 (1), 1–82 (2004).
V. V. Chistyakov, ‘‘The optimal form of selection principles for functions of a real variable,’’ J. Math. Anal. Appl. 310, 609–625 (2005).
V. V. Chistyakov, ‘‘A selection principle for functions with values in a uniform space,’’ Dokl. Math. 74, 559–561 (2006).
V. V. Chistyakov, ‘‘A pointwise selection principle for functions of a real variable with values in a uniform space,’’ Sib. Adv. Math. 16 (3), 15–41 (2006).
V. V. Chistyakov, Metric Modular Spaces: Theory and Applications, Springer Briefs in Mathematics (Springer, Cham, 2015).
V. V. Chistyakov, ‘‘Asymmetric variations of multifunctions with application to functional inclusions,’’ J. Math. Anal. Appl. 478, 421–444 (2019).
V. V. Chistyakov, From Approximate Variation to Pointwise Selection Principles, Springer Briefs in Optimization (Springer, Cham, 2021).
V. V. Chistyakov and S. A. Chistyakova, ‘‘The joint modulus of variation of metric space valued functions and pointwise selection principles,’’ Studia Math. 238 (1), 37–57 (2017).
V. V. Chistyakov and S. A. Chistyakova, ‘‘Pointwise selection theorems for metric space valued bivariate functions,’’ J. Math. Anal. Appl. 452, 970–989 (2017).
V. V. Chistyakov, C. Maniscalco, and Yu. V. Tretyachenko, ‘‘Variants of a selection principle for sequences of regulated and non-regulated functions,’’ in Topics in Classical Analysis and Applications in Honor of Daniel Waterman, Ed. by L. de Carli, K. Kazarian, and M. Milman (World Scientific, Hackensack, NJ, 2008), pp. 45–72.
V. V. Chistyakov and D. Repovš, ‘‘Selections of bounded variation under the excess restrictions,’’ J. Math. Anal. Appl. 331, 873–885 (2007).
J. Dieudonné, Foundations of Modern Analysis (Academic, New York, 1960).
R. M. Dudley and R. Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and p-Variation (Springer, Berlin, 1999).
R. Engelking, General Topology (Heldermann, Berlin, 1989).
D. Fraňková, ‘‘Regulated functions,’’ Math. Bohem. 116 (1), 20–59 (1991).
C. Goffman, G. Moran and D. Waterman, ‘‘The structure of regulated functions,’’ Proc. Am. Math. Soc. 57, 61–65 (1976).
E. Helly, ‘‘Über lineare Funktionaloperationen,’’ Sitzungsber. Naturwiss. Kl. Kaiserl. Akad. Wiss. Wien 121, 265–297 (1912).
T. H. Hildebrandt, Introduction to the Theory of Integration (Academic, New York, London, 1963).
R. L. Jeffery, ‘‘Generalized integrals with respect to functions of bounded variation,’’ Canad. J. Math. 10, 617–628 (1958).
J. L. Kelley, General Topology (Van Nostrand, Princeton, NJ, 1957).
J. Musielak and W. Orlicz, ‘‘On generalized variations (I),’’ Studia Math. 18, 11–41 (1959).
I. P. Natanson, Theory of Functions of a Real Variable, (Nauka, Moscow, 1974; Ungar, New York, 1965).
S. Perlman, ‘‘Functions of generalized variation,’’ Fund. Math. 105, 199–211 (1980).
Š. Schwabik, Generalized Ordinary Differential Equations (World Scientific, River Edge, NJ, 1992).
L. Schwartz, Analyse Mathématique (Hermann, Paris, 1967), Vol. 1.
A. A. Tolstonogov, ‘‘Properties of the space of proper functions,’’ Math. Notes 35, 422–427 (1984).
Yu. V. Tretyachenko, ‘‘A generalization of the Helly theorem for functions with values in a uniform space,’’ Russ. Math. (Iz. VUZ) 54 (5), 35–46 (2010).
S. Willard, General Topology (Addison-Wesley, Reading, MA, 1970).
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Chistyakov, V.V., Chistyakova, S.A. The Approximate Variation of Univariate Uniform Space Valued Functions and Pointwise Selection Principles. Lobachevskii J Math 43, 550–563 (2022). https://doi.org/10.1134/S1995080222060087
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DOI: https://doi.org/10.1134/S1995080222060087