#### For citation:

Lomov I. S. The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2019, vol. 19, iss. 1, pp. 34-58. DOI: 10.18500/1816-9791-2019-19-1-34-58, EDN: XDDPFB

# The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval

The paper discusses the basics of the spectral method of V. A. Il’in on an example of a simple second order differential operator on a segment of the number line. The ﬁrst theorem of Il’in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated. The substantiation of the statements about the uniform convergence of biorthogonal expansions of functions using the Il’in method is presented. The main theorems, including, the recently established theorem for operators with integral boundary conditions are formulated.

- Il’in V. A. The uniform equiconvergence of expansions in the eigen- and associated functions of a nonselfadjoint ordinary differential operator and in a trigonometric Fourier series. Dokl. Akad. Nauk SSSR, 1975, vol. 223, no. 3, pp. 548–551 (in Russian).
- Il’in V. A. On the equiconvergence of expansions in the trigonometric Fourier series and eigenfunctions of the beam M. V. Keldysh non-selfadjoint ordinary differential operator. Dokl. Akad. Nauk SSSR, 1975, vol. 225, no. 3, pp. 497–499 (in Russian).
- Il’in V. A. Spektralnaya teoriya differentsialnykh operatorov. Samosopryazhennye differentsialnye operatory [Spectral theory of differential operators. Selfadjoint differential operators]. Moscow, Nauka, 1991. 368 p. (in Russian).
- Bari N. K. Biorthogonal systems and bases in Hilbert space. Uch. Zap. Mosk. Gos. Univ., 1951, vol. 148, pp. 69–107 (in Russian).
- Il’in V. A. On the unconditional basis property for systems of eigenfunctions and associated functions of a second-order differential operator on a closed interval. Dokl. Akad. Nauk SSSR, 1983, vol. 273, no. 5, pp. 1048–1053 (in Russian).
- Il’in V. A. Necessary and sufficient conditions for the Riesz basis property of root vectors of discontinuous operators of second order. Differ. Uravn., 1986, vol. 22, no. 12, pp. 2059– 2071 (in Russian).
- Il’in V. A. On the basis of Riesz systems of root vector-functions of discontinuous of the Schr¨odinger operator with a matrix potential. Dokl. Akad. Nauk SSSR, 1990, vol. 314, no. 1, pp. 59–62 (in Russian).
- Moiseev E. I., Barnovska M. On the unconditional basis property of a root functions of the first order differential operator in the space of vector-functions. Math. Slovaca, 1990, vol. 40, no. 3, pp. 325–336 (in Russian).
- Kerimov N. B. Some questions of non-self-adjoint differential operator spectral theory : Thesis Diss. Cand. Sci. (Math. and Mech.). Moscow, 1987. 15 p. (in Russian).
- Kerimov N. B. On the unconditional basis property of a system of a root functions of the differential operator of the fourth order. Dokl. Akad. Nauk SSSR, 1986, vol. 286, no. 4, pp. 803–808 (in Russian).
- Kerimov N. B. Basis and uniform minimality of root systems functions of ordinary differential operators : Thesis Diss. Dr. Sci. (Math. and Mech.). Moscow, 1996. 25 p. (in Russian).
- Kritskov L. V. On necessary conditions for a basis in L p(G) of systems of root functions of a one-dimensional Schr¨odinger operator. Dokl. Akad. Nauk SSSR, 1990, vol. 311, no. 6, pp. 1306–1309 (in Russian).
- Il’in V. A. Convergence of eigenfunction expansions at points discontinuities of differential operator coefficients. Math. Notes, 1977, vol. 22, no. 5, pp. 870–882. DOI: https://doi.org/10.1007/BF01098352
- Budaev V. D. On the unconditional basis property of systems of root functions of the second order operator with discontinuous coefficients. Dokl. Akad. Nauk SSSR, 1986, vol. 289, no. 4, pp. 777–780 (in Russian).
- Budaev V. D. On Bessel inequality for systems of root functions differential operators. Dokl. Akad. Nauk SSSR, 1991, vol. 218, no. 1, pp. 16–20 (in Russian).
- Budaev V. D. Criteria of Bessel and Riesz basis of root systems functions of differential operators. Differential Equations, 1991, vol. 27, no. 12, pp. 1421–1432.
- Budaev V. D. Unconditional basis of systems of root functions of ordinary differential operators : Thesis Diss. Dr. Sci. (Math. and Mech.). Moscow, 1993. 22 p. (in Russian).
- Lomov, I. S. The Bessel Inequality, the Riesz theorem and unconditional basis for root vectors of ordinary differential operators. Vestnik Mosk. Un-ta, Ser. 1, Math. Mehan., 1992, no. 5, pp. 33–43 (in Russian).
- Kurbanov V. M. On distribution of eigenvalues and convergence of biorthogonal expansions in root functions of ordinary differential operators : Thesis Diss. Dr. Sci. (Math. and Mech.). Moscow, 2000. 26 p. (in Russian).
- Kritskov L. V. A uniform estimate for the order of associated functions, and the distribution of eigenvalues of a one-dimensional Schr¨odinger operator. Differential Equations, 1989, vol. 25, no. 7, pp. 784–791.
- Kritskov L. V.Some spectral properties of singular ordinary operators of the second order : Thesis Diss. Dr. Sci. (Math. and Mech.). Moscow, 1990. 19 p. (in Russian).
- Il’in V. A., Moiseev E. I. On the systems consisting of subsets of root function of two distinct boundary value problems. Proc. Steklov Inst. Math., 1994, vol. 201, pp. 183–192.
- Kolmogorov A. N., Fomin S. V. Elements of the theory of functions and functional analysis. Moscow, Nauka, 1972. 496 p. (in Russian).
- Krall A. M. The development of general differential boundary systems. Rocky Mountain J. Math., 1975, vol. 5, no. 4, pp. 493–542.
- Dikinov Kh. Zh., Kerefov A. A., Nakhushev A. M. A certain boundary value problem for a loaded heat equation. Differ. Uravn., 1976, vol. 12, no. 1, pp. 177–179 (in Russian).
- Nakhushev A. M. Nagruzhennye uravneniia i ikh primenenie [The loaded equations and their application]. Moscow, Nauka, 2012. 232 p. (in Russian).
- Lomov I. S. Example of discontinuous operator with a discontinuous adjoint operator. Basis property. Zadachi matematicheskoi fiziki i spektral’naia teoriia operatorov [Problems of Mathematical Physics and Spectral Theory of Operators], Collection of Scientific Papers of Moscow Power Engineering Institute, vol. 215. Moscow, Moscow Power Engineering Inst, 1989, pp. 46–50 (in Russian).
- Lomov I. S. The basis property of root functions of operators with multipoint boundary conditions. Differ. Uravn., 1989, vol. 25, no. 6, pp. 1053–1056 (in Russian).
- Lomov I. S. Properties of root functions of the Sturm–Liouville operator that are discontinuous on an everywhere dense set. Soviet Math. (Iz. VUZ), 1990, no. 8, pp. 39–49.
- Lomov I. S. The basis property of root vectors of loaded second-order differential operators on an interval. Differential Equations, 1991, vol. 27, no. 1, pp. 64–75.
- Lomov I. S. A theorem on the unconditional basis property of root vectors of second-order weighted differential operators. Differential Equations, 1991, vol. 27, no. 9, pp. 1098–1107.
- Lomov I. S. On the basis property of systems of nonregular root vectors of higher-order differential operators. Differential Equations, 1993, vol. 29, no. 1, pp. 62–72.
- Belyancev O. V. The bessel inequality and the basis property of root functions of a secondorder singular differential operator. Differential Equations, 2000, vol. 36, no. 8, pp. 1119– 1130. DOI: https://doi.org/10.1007/BF02754179
- Yurko V. A. Boundary value problems with discontinuity conditions in an interior point of the interval. Differential Equations, 2000, vol. 36, no. 8, pp. 1266–1269. DOI: https://doi.org/10.1007/BF02754199
- Makin A. S. Convergence of the Riesz means of spectral expansions that correspond to the one-dimensional Schr¨odinger operator. Differ. Uravn., 1988, vol. 24, no. 5, pp. 897–899 (in Russian).
- Makin A. S. On the average of the Riesz biorthogonal expansions in root functions of non-self-adjoint Schr¨odinger operator extensions. Dokl. Akad. Nauk, 1992, vol. 322, no. 3, pp. 472–475 (in Russian).
- Makin A. S. On the properties of the root functions and spectral decompositions, responding to non-self-adjoint differential operators : Thesis Diss. Dr. Sci. (Math. and Mech.). Moscow, 2000. 26 p. (in Russian).
- Il’in V. A., Tikhomirov V. V. The basis property of Riesz means of spectral decompositions corresponding to an nth-order ordinary nonselfadjoint differential operator. Differ. Uravn., 1982, vol. 18, no. 12, pp. 2098–2126.
- Il’in V. A. Evaluation of the difference between means of two Riesz spectral decompositions for functions of class L 2. Differential Equations, 1988, vol. 24, no. 5, pp. 852–863.
- Salimov Ya. Sh. On average Riesz of expansions in root functions of some nonlocal boundary value problems. Differential Equations, 1987, vol. 23, no. 1, pp. 155–160.
- Tikhomirov V. V. On unconditional basis of root vectors of loaded operators. Differential Equations, 1989, vol. 25, no. 2, pp. 355–357.
- Tikhomirov V. V. On the unconditional basis property of root vectors of nonlocal problems for systems of equations with deviating argument. Differential Equations, 1990, vol. 26, no. 1, pp. 147–53.
- Barnovska M., Tikhomirov V. V. Riesz basis property of root vectors of nonlocal problems for systems of differential equations. Math. Slovaca, 1993, vol. 43, no. 2, pp. 193–205.
- Lomov I. S. Some properties of spectral expansions related to operators of the Sturm– Liouville problem. Dokl. Akad. Nauk SSSR, 1979, vol. 248, no. 5, pp. 1063–1065 (in Russian).
- Lomov I. S. Some properties of eigenfunctions and adjoint functions of operator of the Sturm–Liouville problem. Differential Equations, 1982, vol. 18, no. 10, pp. 1684–1694.
- Tikhomirov V. V. Exact estimates of regular one-dimensional solutions non-self-adjoint Schr¨odinger equation. Dokl. Akad. Nauk SSSR, 1982, vol. 273, no 4, pp. 807–810 (in Russian).
- Tikhomirov V. V. Exact estimates of eigenfunctions of arbitrary non-self-adjoint Schr¨odinger operator. Differential Equations, 1983, vol. 19, no. 8, pp. 1378–1385.
- Lomov I. S. Estimates of root functions of the operator adjoint to a second order differential operator with integral boundary conditions. Differential Equations, 2018, vol. 54, no. 5, pp. 596–607. DOI: https://doi.org/10.1134/S001226611805004X
- Il’in V. A., Kritskov L. V. Properties of Spectral Expansions Corresponding to Non-SelfAdjoint Differential Operators. J. Math. Sci., 2003, vol. 116, iss. 5, pp. 3489–3550. DOI: https://doi.org/10.1023/A:1024180807502
- Khromov A. P. Spectral analysis of differential operators on a finite interval. Differential Equations, 1995, vol. 31, no. 10, pp. 1691–1696.
- Khromov A. P. Equiconvergence theorems for integrodifferential and integral operators. Math. USSR-Sb., 1981, vol. 42, no. 3, pp. 331–355. DOI: https://doi.org/10.1070/SM1982v042n03ABEH002257
- Khromov A. P. On equiconvergence of the eigenfunction finite-dimensional perturbations of the integration operator. Vestn. Moscow State Univ., Ser. 1, Math. Mech., 2000, no. 2, pp. 21–26 (in Russian).
- Kurdyumov V. P., Khromov A. P. Riesz basis formed by root functions of a functionaldifferential equation with a reflection operator. Differential Equations, 2008, vol. 44, no. 2, pp. 203–212.
- Burlutskaya M. S., Khromov A. P. Resolvent approach in the Fourier method. Dokl. Math., 2014, vol. 90, no. 2, pp. 545–548. DOI: https://doi.org/10.1134/S1064562414060076
- Lomov I. S. Integral representations of irregular root functions of loaded second-order differential operators. Differential Equations, 2016, vol. 52, no. 12, pp. 1634–1646. DOI: https://doi.org/10.1134/S0012266116120041
- Shkalikov A. A. On the basis property of eigenfunctions of ordinary differential operator with integral boundary conditions. Vestn. Moscow State Univ., Ser. 1, Math. Mech., 1982, no. 6, pp. 12–21 (in Russian).
- Lomov I. S. Uniform Convergence of Biorthogonal Series for the Schr¨odinger Operator with Multipoint Boundary Conditions. Differential Equations, 2002, vol. 38, no. 7, pp. 941–948. DOI: https://doi.org/10.1023/A:1021147327871
- Gomilko A. M., Radzievskii G. V. Basis properties of eigenfunctions of a regular boundary value problem for the vector functional differential equations. Differential Equations, 1991, vol. 27, no. 3, pp. 264–273.
- Khromov A. P. O ravnoskhodimosti razlozhenij po sobstvennym funkciyam operatora differencirovaniya s integral’nym granichnym usloviem [On equiconvergence of the eigenfunction expansions of the differential operator with integral boundary condition]. Matematika. Mekhanika [Mathematics. Mechanics], Saratov, 2003, vol. 5, pp. 129–131 (in Russian).
- Khromov A. P. On the analogue of Jordan–Dirichlet theorem for expansions in eigenfunctions of differential-difference operator with the integral boundary condition. Proc. of the Academy of Natural Sciences (Volga Inter-Regional Department), 2004, no. 4, pp. 80–87 (in Russian).
- Sedletskii A. M. Approximation properties of systems of exponentials in Lp(a,b). Differential Equations, 1995, vol. 31, no. 10, pp. 1639–1645.
- Pulkina L. S., Dyuzheva A. V. Nonlocal problem with variables in time by Steklov boundary conditions for the hyperbolic equation. Vestnik SamGU, Estestvenno-Nauchnaya Ser., 2010, iss. 4 (85), pp. 56–64 (in Russian).
- Samarskaya T. A. Equiconvergence of spectral expansions that correspond to nonselfadjoint extensions of a second-order differential operator. Differential Equations, 1988, vol. 24, no. 1, pp. 122–131.
- Naimark M. A. Linear differential operators. Moscow, Nauka, 1969. 526 p. (in Russian).
- Il’in V. A., Moiseev E. I. A nonlocal boundary value problem for the Sturm-–Liouville operator in differential and difference interpretations . Dokl. Akad. Nauk SSSR, 1986, vol. 291, no. 3, pp. 534–539 (in Russian).
- Samarskaya T. A. Absolute and uniform convergence of expansions in root functions of a nonlocal boundary value problem of the first kind. Differential Equations, 1989, vol. 25, no. 7, pp. 813–817.
- Mustafin M. A. Absolute and uniform convergence of series in a sine system. Differ. Uravn., 1992, vol. 28, no. 8, pp. 1465–1466 (in Russian).
- Lazetic N. L. On uniform convergence on closed intervals of spectral expansions and their derivatives, for functions from W1 p . Matematicki Vesnik, 2004, vol. 56, no. 3–4, pp. 91–104.
- Kurbanov V. M. Conditions for the absolute and uniform convergence of the biorthogonal series corresponding to a differential operator. Dokl. Math., 2008, vol. 78, no. 2, pp. 748– 750. DOI: https://doi.org/10.1134/S1064562408050281
- Lomov I. S. Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators. Izv. Saratov Univ. (N. S. ), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 4, pp. 405–418 (in Russian). DOI: https://doi.org/10.18500/18169791-2015-15-4-405-418

- 1728 reads