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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 3(363), Pages 97–168 (Mi umn1430)  

This article is cited in 10 scientific papers (total in 10 papers)

Almost periodic functions and representations in locally convex spaces

A. I. Shtern

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Properties of diverse classes of almost periodic functions with values in locally convex spaces and of almost periodic representations on locally convex spaces are considered. The well-known criterion for the almost periodicity of weakly almost periodic group representations on Banach spaces (in terms of scalar almost periodicity) is extended to the case of weakly continuous weakly almost periodic representations on barrelled spaces in which the weakly closed convex hulls of weakly compact sets are weakly compact. Applications of this result are indicated and a survey of the current state of some other classical problems in the theory of almost periodic functions (as applied to almost periodic functions with values in locally convex spaces) and modern directions of investigation related to almost periodic functions on groups and finite-dimensional unitary representations of groups are presented. In particular, decomposition problems for weakly almost periodic representations and characterizations of diverse classes of almost periodic functions (including criteria for almost periodicity), existence problems for the mean value, countability conditions for the spectrum of a scalarly almost periodic function, theorems on the integral and the differences of almost periodic functions, and other relationships among strong, scalar, and weak almost periodicity for functions with values in locally convex spaces are treated.

DOI: https://doi.org/10.4213/rm1430

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English version:
Russian Mathematical Surveys, 2005, 60:3, 489–557

Bibliographic databases:

UDC: 517.986.63+517.986.4
MSC: Primary 43A60, 22A25; Secondary 42A75, 43A07, 22A20, 46A32, 47D03, 46A08, 22D10, 3
Received: 18.08.2004

Citation: A. I. Shtern, “Almost periodic functions and representations in locally convex spaces”, Uspekhi Mat. Nauk, 60:3(363) (2005), 97–168; Russian Math. Surveys, 60:3 (2005), 489–557

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751  mathnet  crossref  mathscinet  zmath  elib  elib
    2. A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. I. Shtern, “Duality between compactness and discreteness beyond Pontryagin duality”, Proc. Steklov Inst. Math., 271 (2010), 212–227  mathnet  crossref  mathscinet  isi  elib
    4. A. I. Shtern, “The structure of homomorphisms of connected locally compact groups into compact groups”, Izv. Math., 75:6 (2011), 1279–1304  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. M. I. Karakhanian, “Almost periodicity in spectral analysis representations induced by generalized shift operation”, Uch. zapiski EGU, ser. Fizika i Matematika, 2012, no. 3, 9–13  mathnet
    6. Khadjiev D., Cavus A., “Continuous Invariant Averagings”, Turk. J. Math., 37:5 (2013), 770–780  crossref  mathscinet  zmath  isi  elib
    7. I. A. Trishina, “Pochti periodicheskie na beskonechnosti funktsii otnositelno podprostranstva integralno ubyvayuschikh na beskonechnosti funktsii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:4 (2017), 402–418  mathnet  crossref  elib
    8. Shtern I A., “Continuity Conditions For Finite-Dimensional Locally Bounded Representations of Connected Locally Compact Groups”, Russ. J. Math. Phys., 25:3 (2018), 345–382  crossref  mathscinet  zmath  isi  scopus
    9. A. G. Baskakov, V. E. Strukov, I. I. Strukova, “Harmonic analysis of functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity”, Sb. Math., 210:10 (2019), 1380–1427  mathnet  crossref  crossref  adsnasa  isi  elib
    10. Kostic M., “Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations”, Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations, Walter de Gruyter Gmbh, 2019, 1–329  isi
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