BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , ON A CRITERION FOR THE CONTINUITY OF THE RAYLEIGH–RITZ OPERATOR // BSU Bulletin. Mathematics, Informatics. - 2018. №3. . - С. 3-13.
Title:
ON A CRITERION FOR THE CONTINUITY OF THE RAYLEIGH–RITZ OPERATOR
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Annotation:
The work considers the Rayleigh–Ritz operator identified on the set of pairs of measurable functions that equals to the ratio of their modules if a denominator is different from zero, and zero otherwise. The issue of the continuity of this operator regarding the convergence in measure is studied. It is shown that for the conver- gence of the value of operator on a sequence of pairs to the value on the limit pair of functions, it is necessary not only the convergence in measure of its arguments, but also the convergence in measure of the second argument to the carrier of its limit.
Keywords:
measure; -algebra; convergence in measure; topology; the Rayleigh–Ritz operator; carrier of function; metric; characteristic function.
List of references:
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Dyachenko M. I., Ulyanov P. L. Mera i integral [Measure and Integral]. Mos- cow: Faktorial Publ., 1998. 160 p.
Daneev A. V., Rusanov V. A., Sharpinskii D. Yu. Nestacionarnaja realizacija Kalmana-Mesarovicha v konstrukcijah operatora Releja-Ritca [The Nonstationary Re- alization of Kalman-Mesarovic in the Constructions of the Rayleigh-Ritz Operator] Kibernetika i sistemnyi analiz - Cybernetics and Systems Analysis. 2007. No. 1. Pp. 82–90.
Lakeev A. V., Linke Yu. E, Rusanov V. A. K realizacii polilinejnogo regulja- tora differencial'noj sistemy vtorogo porjadka v gil'bertovom prostranstve [To Realization of a Polylinear Controller as a Second-Order Differential System in a Hil- bert Space]. Differentsial'nye uravneniya - Differential Equations. 2017. V. 53, No. 8. Pp. 1070–1081. DOI: 10.1134/S0012266117080122.
Rusanov V. A., Daneev A. V., Linke Yu. É. K geometricheskim osnovam differ- entsial'noi realizatsii dinamicheskikh protsessov v gil'bertovom prostranstve [To the Geometric Foundations of the Differential Realization of Dynamical Processes in a Hilbert Space]. Kibernetika i sistemnyi analiz - Cybernetics and Systems Analysis. 2017. V. 53, No. 4. Pp. 554564. DOI: 10.1007/s10559-017-9957-z.
Kantorovich L. V., Akilov G. P. Funkcionalnyj Analiz [Functional Analysis]. Moscow: Nauka Publ., 1984. 752 p.
Dyachenko M. I., Ulyanov P. L. Mera i integral [Measure and Integral]. Mos- cow: Faktorial Publ., 1998. 160 p.
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