01.01.01 (Real analysis, complex analysis, and functional analysis)
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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.
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.
Biography
Graduated from Faculty of Mathematics and Mechanics of Saratov State University (SSU) in 1976 (department of differential equations and applyed mathematics). Ph.D. thesis was defended in 1981. A list of my works contains more than 60 titles.
Main publications:
V. S. Rykhlov. Asymptotical formulas for solutions of linear differential systems of the first order // Results in Mathematics, v. 36, no. 3–4, 1999, p. 342–353.
G. Freiling, V. Rykhlov. Pointwise convergence of eigenfunctions for a general class of regular eigenvalue problems // Methods of Functional Analysis and Topology, 1997, v. 3, p. 27–45.
V. S. Rykhlov. Equiconvergence rate in terms of general moduli of continuity for differential operators // Results in Mathematics, v. 29, no. 1, 1996, p. 153–168.
V. S. Rykhlov, “Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 232 (2024), 99–121
2023
2.
V. S. Rykhlov, “Generalized initial-boundary problem for the wave equation with mixed derivative”, CMFD, 69:2 (2023), 342–363
V. S. Rykhlov, “On the solution of the initial-boundary problem in a half-strip for a hyperbolic equation with a mixed derivative”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 226 (2023), 89–107
V. S. Rykhlov, “The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution”, Izv. Saratov Univ. Math. Mech. Inform., 23:2 (2023), 183–194
V. S. Rykhlov, “Generalized solution of the simplest initial boundary value problem for a homogeneous hyperbolic equation with a mixed derivative”, Taurida Journal of Computer Science Theory and Mathematics, 2023, no. 2, 72–88
2022
6.
V. S. Rykhlov, “On the completeness of eigenfunctions of one $5$th-order differential operator”, CMFD, 68:2 (2022), 338–375
7.
V. S. Rykhlov, “Solvability of a mixed problem for a hyperbolic equation with splitting boundary conditions in the case of incomplete system of eigenfunctions”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 204 (2022), 124–134
V. S. Rykhlov, “Solvability of the mixed problem for a hyperbolic equation in the case of incomplete system of eigenfunctions”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 200 (2021), 95–104
V. S. Rykhlov, “Multiple completeness of the root functions of the pencils of differential operators with constant coefficients and splitting boundary conditions”, Izv. Saratov Univ. Math. Mech. Inform., 19:2 (2019), 134–151
2018
10.
V. S. Rykhlov, “On multiple completeness of the root functions of the nonregular pencils of differential operators with constant coefficients and splitting boundary conditions”, Taurida Journal of Computer Science Theory and Mathematics, 2018, no. 4, 90–112
2017
11.
V. S. Rykhlov, “On multiple completeness of the root functions of ordinary differential polynomial pencil with constant coefficients”, CMFD, 63:2 (2017), 340–361
V. S. Rykhlov, “Expansion in root functions of strongly irregular pencil of differential operators of the second order with multiple characteristics”, Izv. Saratov Univ. Math. Mech. Inform., 16:2 (2016), 165–174
2014
13.
V. S. Rykhlov, O. V. Blinkova, “On Multiple Completeness of the Root Functions of a Certain Class of Pencils of Differential Operators with Constant Coefficients”, Izv. Saratov Univ. Math. Mech. Inform., 14:4(2) (2014), 574–584
V. S. Rykhlov, “Expansion in Eigenfunctions of Quadratic Strongly Irregular Pencils of Differential Operators of the Second Order”, Izv. Saratov Univ. Math. Mech. Inform., 13:1(1) (2013), 21–26
V. S. Rykhlov, O. V. Parfilova, “On multiple completeness of the root functions of the pencils of differential operators with constant coefficients”, Izv. Saratov Univ. Math. Mech. Inform., 11:4 (2011), 45–58
V. S. Rykhlov, “On multiple completeness of the root functions for a class of the pencils of differential operators”, Izv. Saratov Univ. Math. Mech. Inform., 10:2 (2010), 24–34
V. S. Rykhlov, “On properties of the eigenfunctions of a quadratic pencil of the second order differential operators”, Izv. Saratov Univ. Math. Mech. Inform., 9:1 (2009), 31–44
V. S. Rykhlov, “Completeness of eigenfunctions of one class of pencils of differential operators with constant coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 6, 42–53; Russian Math. (Iz. VUZ), 53:6 (2009), 33–43
V. S. Rykhlov, “The rate of equiconvergence for differential operators with a nonzero coefficient multiplying the $(n-1)$st derivative”, Differ. Uravn., 26:6 (1990), 975–989; Differ. Equ., 26:6 (1990), 704–715
V. S. Rykhlov, “The rate of equiconvergence for differential operators with nonzero coefficient multiplying the $(n-1)$th derivative”, Dokl. Akad. Nauk SSSR, 279:5 (1984), 1053–1056
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