ON A CLASS OF NONSTATIONARY CURVES IN HILBERT SPACE
DOI:
https://doi.org/10.20998/2079-0023.2023.02.15Keywords:
Hilbert space, nonstationary random processes, correlational function, infinitesimal function, conjugate operator, dissipative operator, operator spectrum, dissipative spectrum, contrast free spectrumAbstract
Stationary random processes have been studied quite well over recent years starting with the works of A. N. Kolmogorov. The possibility of building nonstationary random process correlation theory was considered in the monographs by M. S. Livshits, A. A. Yantsevich, V. A. Zolotarev and others. Some classes of nonstationary curves were investigated by V. E. Katsnelson. In this paper nonstationary random processes are represented as curves in Hilbert space which "slightly deviate" from random processes with the correlation function of special kind. The infinitesimal correlation function has been introduced; in essence, this function characterizes the deviation from the correlation process with the given correlation function. The paper discusses the cases of nonstationary random processes, the operator of which has one‑dimensional imaginary component. Cases of a dissipative operator with descrete spectrum are also considered in this work. It is shown that the nonstationarity of the random process is closely related to the deviation of the operator from its conjugated operator. Using the triangle and universal models of non‑self‑ajoint operators it is possible to obtain the representation for the correlation function in the case of nonstationary process which replaces the Bochner – Khinchin representation for stationary random processes. The expresson for the infinitesimal correlation function was obtained for different cases of operator spectrum: for the descrete spectrum placed in the upper half‑plane and for the contrast‑free spectrum at zero. In the case of dissipative operator with descrete spectrum the infinitesimal function can be found in terms of special lambda function. For Lebesque spaces of compex‑valued squared integratable functions the expresson of infinitesimal function was found in terms of special zero order modified Bessel function. It was shown that a similar approach can be applied for the evolutionarily represented sequences in Hilbert spaces.
References
Livshitz M. S., Yantsevich A. A. Operator colligations in Hilbert spaces. New-York, John Wiley and Sons, 1979. 226 p.
Akhiyezer N. I., Glazman I. M. Teoriya lineynykh operatorov v gil'bertovom prostranstve [The theory of linear operators in Hilbert space]. Kharkov, Vischa Shkola Publ., 1978, vol. 2. 300 p.
Kolmogorov A. N., Fomin S. V. Elementi teoriy funktsiy i funktsional'nogo analiza [Elements of function theory and functional analysis]. Moscow, Nauka Publ., 1968. 492 p.
Riesz F. , Sz. – Nagy B. Functional analysis. 6th Revised ed. Budapest, 1972, 587 p. (Russ. ed.: Riesz F. , Sz. – Nagy B. Lektsii po functional'nomu analizu. Moscow, Mir Publ., 1979. 587 p.).
Kolmogorov A. N. Stationary sequences in Hilbert spaces. Buyll. Mosk. Gos. Univ. 1941, vol. 2, no. 6, pp. 1–40.
Zolotarev V. A., Yantsevich A. A. Nestatsionarnyye krivyye v gil'bertovom prostranstve i nelineyynyye operatornyye uravneniya [The nonstationary curves in Hilbert space and nonlinear operator equations]. Teoriya operatorov subgarmonicheskikh funksiy [The theory of the operators of subharmonic functions]. Kyiv, Naukova dumka Publ., 1991, pp. 54–59.
Yantsevich A. A. Nonstationary sequences in Hilbert space. Correlation Theory. Journal of. Soviet Mathematics. 1990, vol. 48, no. 4, pp. 440–443.
Yantsevich A. A. The application of operator colligatioins to the investigation of nonstationary random processes and sequences. Materials of the All Union Simposium on Random Processes. Kyiv, 1973, pp. 229–232.
Petrova A. Yu. Korrelyatsyonnaya teoriya nekotorikh klassov sluchaynykh funksiy konechnogo ranga nestatsionarnosti [The correlation theory of some classes of functions of final rank of nonstationarity]. Radioelektronika i Informatika. 2007, vol. 1, pp. 67–71.
Katsnelson V. E. Ob usloviyakh bazisnosti sistemy kornevykh vektorov nekotorikh klassov operatorov [On the base conditions of the system root vektors of some classes of operators]. Functional'nyy analiz i yego prilozheniya. 1967, vol. 1, issue 2, pp. 39–51.
Hatamleh R. On a class of nonhomogeneous fields in Hilbert space. Journal of Mathematics and Statistics. 2007, vol. 3, no. 1, pp. 207–210.
Zolotarev V. A. Analiticheskiye metody spektral'nykh predstavleniy nesamosopryadzennykh i neunitarnykh operatorov [The analytical methods of spectral representation of nonselfajoint and nonunitary operators]. Kharkov, KhNU Publ., 2003, 342 p.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
