Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Phone: +7 (8452) 26 46 94, +7 (8452) 22 36 64
E-mail: Keywords: ordinary differential operator,
eigenvalues,
eigenfunctions and associated functions,
root functions,
expansion in biorthogonal series,
Riesz basis property,
completeness of the system eigenfunction and associated functions,
pencil of ordinary differential operators,
multiple completeness of the system of eigenfunctions and associated functions,
Green's function,
regular boundary conditions,
nonregular boundary conditions. expansion by root functions,
equivalence of decompositions,
estimation of the difference of partial sums of decompositions,
eigenvalues and attached functions,
multiple completeness of root functions,
asymptotic of the system of solutions of differential equations,
spectral parameter.
UDC: 517.927, 517.928, 517.929, 517.984, 517.984.55, 517.984.6, 517.927.25 MSC: 34B05, 34B09, 34B10, 34B24, 34B27, 34L05, 34L10, 34L20, 47E05
Subject:
An asympthotic of the fundamental system of solutions of the differential equation generated by ordinary linear differential expression of the $n$th order with a spectral parameter and nonsmooth coefficient at $n-1$st derivative was built. For the ordinary differential operator, generated by this differential expression and regular two-point boundary conditions, theorems of equiconvergence of the expansions of an arbitrary function in biorthogonal series with respect to the eigen- and associated functions of this operator and in the ordinary trigonometric Fourier series were proved. Estimates of the rate of equiconvergence were given. Conditions of completeness and Riesz basisness of the system of eigen- and associated functions of this operator in the space of summable with square functions were found.
Main publications:
\begin{thebibliography}{99}
\RBibitem{1}
\by V. S. Rykhlov
\paper Asymptotical formulas for solutions of linear differential systems of the first order
\jour Results in Mathematics,
\vol 36
\issue 3--4
\yr 1999
\pages 342--353
\RBibitem{1}
\by G. Freiling, V. Rykhlov
\paper Pointwise convergence of eigenfunctions for a general class of regular eigenvalue problems
\jour Methods of Functional Analysis and Topology
\yr 1997
\vol 3
\pages 27--45
\RBibitem{1}
\by V. S. Rykhlov
\paper Equiconvergence rate in terms of general moduli of continuity for differential operators
\jour Results in Mathematics
\vol 29
\issue 1
\yr 1996
\pages 153--168
\Bibitem{Ryk17}
\by Rykhlov V. S.
\paper Multiple completeness of the root functions for a certain class of pencils of ordinary differential operators with constant coefficient
\jour Results in mathematics
\yr 2017
\vol 72
\issue 1-2
\pages 281-301
\Bibitem{Ryk99}
\by Rykhlov V. S.
\paper Asymptotical formulas for solutions of linear differential systems of the first order
\jour Results in mathematics
\yr 1999
\vol 36
\pages 342-353
\Bibitem{1}
\by Rykhlov V. S.
\paper Multiple completeness of the root functions for a certain class of ordinary differential pencils with constant coefficients
\jour Lobachevskii J. Math.
\yr 2017
\vol 38
\issue 4
\pages 730-740
