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Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 6, Pages 1348–1387 (Mi izv2073)  

This article is cited in 6 scientific papers (total in 6 papers)

On the Dirichlet problem for a pseudodifferential equation encountered in the theory of random processes

B. V. Pal'tsev


Abstract: The problem is considered of finding a function $u(t)$ satisfying the equation
\begin{equation} \mathscr F^{-1}[\tilde k(x)\tilde u(x)](t)=f(t)\quadfor\quad t\in\Omega,\qquad\tilde u(x)=\mathscr F[u(t)](x), \end{equation}
and the conditions
\begin{equation} u(t)\equiv0\quadfor\quad t\notin\Omega,\qquad\int_{-\infty}^{+\infty}\tilde k(x)|\tilde u(x)|^2 dx<\infty, \end{equation}
where $\tilde k(x)$ is a nonnegative measurable function and $\mathscr F$ is the Fourier operator. An existence and uniqueness theorem is proved under quite general assumptions concerning the spectral densities $\tilde k(x)$. Explicit formulas for the solution of problem (1), (2) are obtained in the case when $\Omega$ is an interval $(-T,T)$ and $\tilde k(x)=|x|^\alpha$, $\alpha>0$.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:6, 1285–1322

Bibliographic databases:

UDC: 517.9
MSC: Primary 35S15; Secondary 60G25, 62M20
Received: 23.09.1976

Citation: B. V. Pal'tsev, “On the Dirichlet problem for a pseudodifferential equation encountered in the theory of random processes”, Izv. Akad. Nauk SSSR Ser. Mat., 41:6 (1977), 1348–1387; Math. USSR-Izv., 11:6 (1977), 1285–1322

Citation in format AMSBIB
\Bibitem{Pal77}
\by B.~V.~Pal'tsev
\paper On the Dirichlet problem for a~pseudodifferential equation encountered in the theory of random processes
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 6
\pages 1348--1387
\mathnet{http://mi.mathnet.ru/izv2073}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=499862}
\zmath{https://zbmath.org/?q=an:0372.35074|0396.35089}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 6
\pages 1285--1322
\crossref{https://doi.org/10.1070/IM1977v011n06ABEH001769}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. B. V. Pal'tsev, “A generalization of the Wiener–Hopf method for convolution equations on a finite interval with symbols having power-like asymptotics at infinity”, Math. USSR-Sb., 41:3 (1982), 289–328  mathnet  crossref  mathscinet  zmath
    2. B. V. Pal'tsev, “Convolution equations on a finite interval for a class of symbols having powerlike asymptotics at infinity”, Math. USSR-Izv., 16:2 (1981), 291–356  mathnet  crossref  mathscinet  zmath  isi
    3. B. V. Pal'tsev, “A method for constructing a canonical matrix of solutions of a Hilbert problem arising in the solution of convolution equations on a finite interval”, Math. USSR-Izv., 19:3 (1982), 559–610  mathnet  crossref  mathscinet  zmath
    4. Yu. I. Karlovich, I. M. Spitkovsky, “Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type”, Math. USSR-Izv., 34:2 (1990), 281–316  mathnet  crossref  mathscinet  zmath
    5. B. V. Pal'tsev, “Asymptotic behaviour of the spectra of integral convolution operators on a finite interval with homogeneous polar kernels”, Izv. Math., 67:4 (2003), 695–779  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. 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
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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