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

This article is cited in 7 scientific papers (total in 7 papers)

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
Received: 23.05.2002

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
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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. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119  mathnet  crossref  mathscinet  adsnasa  isi  elib
    2. S. I. Bezrodnykh, V. I. Vlasov, “Singular Riemann–Hilbert problem in complex-shaped domains”, Comput. Math. Math. Phys., 54:12 (2014), 1826–1875  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  zmath  elib
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    6. Chigansky P. Kleptsyna M., “Exact Asymptotics in Eigenproblems For Fractional Brownian Covariance Operators”, Stoch. Process. Their Appl., 128:6 (2018), 2007–2059  crossref  mathscinet  isi  scopus
    7. Chigansky P., Kleptsyna M., Marushkevych D., “Mixed Fractional Brownian Motion: a Spectral Take”, J. Math. Anal. Appl., 482:2 (2020), 123558  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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