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Mat. Zametki, 2012, Volume 92, Issue 5, Pages 643–661 (Mi mz8963)  

This article is cited in 22 scientific papers (total in 22 papers)

Beurlings theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations

A. G. Baskakov, N. S. Kaluzhina

Voronezh State University

Abstract: The results of the paper are obtained for functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of the Beurling spectrum, or essential spectrum, of functions is introduced. If a continuous unitary character is an essential point of the spectrum of a function, then it is the $\mathrm{c}$-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and the properties of such functions are considered. For a parabolic equation with initial function from a homogeneous space, it is proved that the weak solution as a function of the first argument is a slowly varying function at infinity.

Keywords: Beurling spectrum of a function, locally compact Abelian group, parabolic equation, continuous unitary character, Banach space, Fourier transform, Banach module, directed set, Stepanov set.

DOI: https://doi.org/10.4213/mzm8963

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English version:
Mathematical Notes, 2012, 92:5, 587–605

Bibliographic databases:

UDC: 517.9
Received: 28.10.2010
Revised: 08.06.2011

Citation: A. G. Baskakov, N. S. Kaluzhina, “Beurlings theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations”, Mat. Zametki, 92:5 (2012), 643–661; Math. Notes, 92:5 (2012), 587–605

Citation in format AMSBIB
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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. N. S. Kaluzhina, “Kachestvennye svoistva slabykh reshenii zadachi Koshi”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(1) (2013), 8–13  mathnet  crossref
    2. E. E. Dikarev, “On Bernstein inequality for vectors in Banach spaces”, Ufa Math. J., 5:4 (2013), 75–81  mathnet  crossref  elib
    3. V. I. Kuznetsova, V. G. Kurbatov, “Ob obratimosti raznostno-integralnogo operatora v prostranstve medlenno menyayuschikhsya funktsii”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2013, no. 2, 219–229  elib
    4. A. G. Baskakov, N. S. Kaluzhina, D. M. Polyakov, “Slowly varying on infinity semigroups of operators”, Russian Math. (Iz. VUZ), 58:7 (2014), 1–10  mathnet  crossref
    5. A. G. Baskakov, “Harmonic and Spectral Analysis of Power Bounded Operators and Bounded Semigroups of Operators on Banach Spaces”, Math. Notes, 97:2 (2015), 164–178  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. A. A. Ryzhkova, I. A. Trishina, “O pochti periodicheskikh na beskonechnosti resheniyakh raznostnykh uravnenii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:1 (2015), 45–49  mathnet  crossref  elib
    7. M. S. Bichegkuev, “Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients”, Math. Notes, 99:1 (2016), 24–36  mathnet  crossref  crossref  mathscinet  isi  elib
    8. I. I. Strukova, “On Wiener's Theorem for functions periodic at infinity”, Siberian Math. J., 57:1 (2016), 145–154  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. G. Baskakov, I. A. Krishtal, “Spectral analysis of abstract parabolic operators in homogeneous function spaces”, Mediterr. J. Math., 13:5 (2016), 2443–2462  crossref  mathscinet  zmath  isi  scopus
    10. A. G. Baskakov, L. Yu. Kabantsova, I. D. Kostrub, T. I. Smagina, “Linear differential operators and operator matrices of the second order”, Differ. Equ., 53:1 (2017), 8–17  crossref  mathscinet  zmath  isi  scopus
    11. A. A. Ryzhkova, “Garmonicheskii analiz periodicheskikh na beskonechnosti posledovatelnostei”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 1(38), 22–32  mathnet  crossref
    12. A. G. Baskakov, T. K. Katsaran, T. I. Smagina, “Linear differential second-order equations in Banach space and splitting of operators”, Russian Math. (Iz. VUZ), 61:10 (2017), 32–43  mathnet  crossref  isi
    13. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v odnorodnykh prostranstvakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 2(39), 29–38  mathnet  crossref
    14. 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
    15. A. G. Baskakov, I. A. Krishtal, “Spectral analysis of abstract parabolic operators in homogeneous function spaces, II”, Mediterr. J. Math., 14:4 (2017), UNSP 181  crossref  mathscinet  isi  scopus
    16. A. Baskakov. V. Obukhovskii, P. Zecca, “Almost periodic solutions at infinity of differential equations and inclusions”, J. Math. Anal. Appl., 462:1 (2018), 747–763  crossref  mathscinet  zmath  isi  scopus
    17. A. G. Baskakov, I. I. Strukova, I. A. Trishina, “Solutions almost periodic at infinity to differential equations with unbounded operator coefficients”, Siberian Math. J., 59:2 (2018), 231–242  mathnet  crossref  crossref  isi  elib
    18. A. G. Baskakov, V. E. Strukov, I. I. Strukova, “On the almost periodic at infinity functions from homogeneous spaces”, Probl. anal. Issues Anal., 7(25):2 (2018), 3–19  mathnet  crossref  elib
    19. A. G. Baskakov, E. E. Dikarev, “Spectral theory of functions in studying partial differential operators”, Ufa Math. J., 11:1 (2019), 3–18  mathnet  crossref  isi
    20. V. E. Strukov, I. I. Strukova, “Garmonicheskii analiz medlenno menyayuschikhsya na beskonechnosti polugrupp operatorov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 19:2 (2019), 152–163  mathnet  crossref  elib
    21. 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
    22. I. A. Vysotskaya, “Pochti periodicheskie na beskonechnosti resheniya raznostnykh uravnenii”, Materialy Voronezhskoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy». 28 yanvarya–2 fevralya 2019 g.  Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 38–46  mathnet  crossref  elib
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