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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 4, Pages 67–154 (Mi izv443)

Asymptotic behaviour of the spectra of integral convolution operators on a finite interval with homogeneous polar kernels

B. V. Pal'tsev

Dorodnitsyn Computing Centre of the Russian Academy of Sciences

Abstract: We obtain asymptotic formulae for the eigenvalues of integral convolution operators on a finite interval with homogeneous polar (complex) kernels. In the Fourier–Laplace images, the eigenvalue and eigenfunction problems are reduced to the Hilbert linear conjugation problem for a holomorphic vector-valued function with two components. This problem is in turn reduced to a system of integral equations on the half-line, and analytic properties of solutions of this system are studied in the Mellin images in Banach spaces of holomorphic functions with fixed poles. We study the structure of the canonical matrix of solutions of this Hilbert problem at the singular points, along with its asymptotic behaviour for large values of the reduced spectral parameter. The investigation of the resulting characteristic equations yields three terms (four in the positive self-adjoint case) of the asymptotic expansions of the eigenvalues, along with estimates of the remainders.

DOI: https://doi.org/10.4213/im443

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English version:
Izvestiya: Mathematics, 2003, 67:4, 695–779

Bibliographic databases:

UDC: 517.948.32+35
MSC: 45E10, 45C05, 30E25

Citation: B. V. Pal'tsev, “Asymptotic behaviour of the spectra of integral convolution operators on a finite interval with homogeneous polar kernels”, Izv. RAN. Ser. Mat., 67:4 (2003), 67–154; Izv. Math., 67:4 (2003), 695–779

Citation in format AMSBIB
\Bibitem{Pal03} \by B.~V.~Pal'tsev \paper Asymptotic behaviour of the spectra of integral convolution operators on a~finite interval with homogeneous polar kernels \jour Izv. RAN. Ser. Mat. \yr 2003 \vol 67 \issue 4 \pages 67--154 \mathnet{http://mi.mathnet.ru/izv443} \crossref{https://doi.org/10.4213/im443} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2001765} \zmath{https://zbmath.org/?q=an:1068.45002} \transl \jour Izv. Math. \yr 2003 \vol 67 \issue 4 \pages 695--779 \crossref{https://doi.org/10.1070/IM2003v067n04ABEH000443} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000186267800003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33748676693} 

• http://mi.mathnet.ru/eng/izv443
• https://doi.org/10.4213/im443
• http://mi.mathnet.ru/eng/izv/v67/i4/p67

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This publication is cited in the following articles:
1. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119
2. S. I. Bezrodnykh, V. I. Vlasov, “Singular Riemann–Hilbert problem in complex-shaped domains”, Comput. Math. Math. Phys., 54:12 (2014), 1826–1875
3. A. G. Barsegyan, “O reshenii uravneniya svertki s summarno-raznostnym yadrom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:4 (2015), 613–623
4. Polosin A.A., “On Eigenfunctions of a Convolution Operator on a Finite Interval For Which the Fourier Image of the Kernel Is the Characteristic Function”, Dokl. Math., 96:1 (2017), 389–392
5. Polosin A.A., “Spectrum and Eigenfunctions of the Convolution Operator on a Finite Interval With Kernel Whose Transform Is a Characteristic Function”, Differ. Equ., 53:9 (2017), 1145–1159
6. Chigansky P. Kleptsyna M., “Exact Asymptotics in Eigenproblems For Fractional Brownian Covariance Operators”, Stoch. Process. Their Appl., 128:6 (2018), 2007–2059
7. Chigansky P., Kleptsyna M., Marushkevych D., “Mixed Fractional Brownian Motion: a Spectral Take”, J. Math. Anal. Appl., 482:2 (2020), 123558
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