Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

Bari, N.K., Biorthogonal systems and bases in a Hilbert space, Matematika (Mathematics), Moscow, 1951, vol. 4: Uch. Zap. Moskov. Gos. Univ., no. 148, pp. 69–107.

Kessel’man, G.M., On unconditional convergence of decompositions in eigenfunctions of some differential operators, Izv. Vyssh. Uchebn. Zaved. Mat., 1964, no. 2, pp. 82–93.

Mikhailov, V.P., On Riesz bases in L2(0, 1), Dokl. Akad. Nauk SSSR, 1962, vol. 144, no. 5, pp. 981–984.

Shkalikov, A.A., On the basis problem of the eigenfunctions of an ordinary differential operator, Russ. Math. Surveys, 1979, vol. 34, no. 5 (209), pp. 249–250.

Makin, A.S., On many-point spectral boundary value problems, Differ. Equations, 2000, vol. 36, no. 10, pp. 1461–1468.

Makin, A.S., On the basis property of systems of root functions of regular boundary value problems for the Sturm–Liouville operator, Differ. Equations, 2006, vol. 42, no. 12, pp. 1717–1728.

Makin, A.S., Inverse problems of spectral analysis for the Sturm–Liouville operator with regular boundary conditions. I, II, Differ. Equations, 2007, vol. 43, no. 10, pp. 1364–1375; 2007, vol. 43, no. 12, pp. 1668–1678.

Veliev, O.A. and Shkalikov, A.A., On the Riesz basis property of the eigen-and associated functions of periodic and antiperiodic Sturm–Liouville problems, Math. Notes, 2009, vol. 85, no. 5, pp. 647–660.

Il’in, V.A., On conditions for the basis property of root functions of a non-self-adjoint differential operator, in Izbrannye voprosy matematiki, mekhaniki i ikh prilozhenii (Selected Problems of Mathematics, Mechanics, and Their Applications), Moscow: Mosk. Gos. Univ., 1999, pp. 223–229.

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, Sov. Math. Dokl., 1983, vol. 28, pp. 743–747.

Il’in, V.A., Necessary and sufficient conditions for the eigenvectors and associated vectors of discontinuous second order operators to be Riesz bases, Differ. Equations, 1986, vol. 22, no. 12, pp. 1409–1419.

Kerimov, N.B., On necessary and sufficient conditions for the basis property of systems of root functions of a differential operator, Differ. Equations, 1996, vol. 32, no. 1, pp. 38–45.

Kerimov, N.B., On the unconditional basis property of the system of eigenfunctions and associated functions of a fourth-order differential operator, Sov. Math. Dokl., 1986, vol. 33, pp. 185–189.

Budaev, V.D., A criterion for the unconditional basis property of systems of eigenfunctions and associated functions of ordinary differential operators, Sov. Math. Dokl., 1990, vol. 42, no. 2, pp. 252–255.

Budaev, V.D., Criteria for the Bessel property and the Riesz basis property of systems of root functions of differential operators. I, II, Differ. Equations, 1991, vol. 27, no. 12, pp. 1421–1432; 1992, vol. 28, no. 1, pp. 21–30.

Lomov, I.S., A theorem on the unconditional basis property of the root vectors of loaded second-order differential operators, Differ. Equations, 1991, vol. 27, no. 9, pp. 1098–1107.

Lomov, I.S., On the basis of systems of irregular root vectors of high-order differential operators, Differ. Equations, 1993, vol. 29, no. 1, pp. 62–72.

Kritskov, L.V. and Sarsenbi, A.M., Riesz basis property of system of root functions of second-order differential operator with involution, Differ. Equations, 2017, vol. 53, no. 1, pp. 33–46.

Atkinson, F.V. and Fulton, C.T., Asymptotics of Sturm–Liouville eigenvalues for problems on a finite interval with one limit-circle singularity. I, Proc. Royal Soc. Edinburgh, 1984, vol. 99A, pp. 51–70.

Eberhard, W., Freiling, G., and Zettl, A., Sturm–Liouville problems with singular non-selfadjoint boundary conditions, Math. Nachr., 2005, vol. 278, no. 12–13, pp. 1509–1523.

Freiling, G. and Yurko, V., Boundary value problems with regular singularities and singular boundary conditions, Int. J. Math. Math. Sci., 2005, vol. 2005, no. 9, pp. 1481–1495.

Kritskov, L.V., On the unconditional basis property of systems of root functions of the one-dimensional singular Schrödinger operator, Differ. Uravn., 1991, vol. 27, no. 8, pp. 1446–1447.

Belyantsev, O.V., The Bessel inequality and the basis property of root functions of a second-order singular differential operator, Differ. Equations, 2000, vol. 36, no. 8, pp. 1119–1130.

Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with singular potentials, Math. Notes, 1999, vol. 66, no. 6, pp. 741–753.

Savchuk, A.M., On the eigenvalues and eigenfunctions of the Sturm–Liouville operator with a singular potential, Math. Notes, 2001, vol. 69, no. 2, pp. 245–252.

Vladykina, V.E. and Shkalikov, A.A., Asymptotics of the solutions of the Sturm–Liouville equation with singular coefficients, Math. Notes, 2015, vol. 98, no. 6, pp. 891–899.

Savchuk, A.M. and Shkalikov, A.A., Sturm–Liouville operators with distribution potentials, Tr. Mosk. Mat. Obs., 2003, vol. 64, pp. 143–192.

Mirzoev, K.A. and Shkalikov, A.A., Differential operators of even order with distribution coefficients, Math. Notes, 2016, vol. 99, no. 5, pp. 779–784.

Kritskov, L.V., Estimates for root functions of a singular second-order differential operator, in Functional Analysis in Interdisciplinary Applications. FAIA, Astana, Kazakhstan, October 2017, Kalmenov T.S., Nursultanov E.D., Ruzhansky M.V., and Sadybekov M.A., Eds., Springer Proc. Math. Statistics, vol. 216, Cham, 2017, pp. 1–13.

Il’in, V.A. and Kritskov, L.V., Properties of spectral decompositions corresponding to nonself-adjoint differential operators, in Itogi Nauki Tekh. Ser. Funkts. Anal. Sovr. Mat. Prilozhen., Moscow, 2006, vol. 96, pp. 190–231.

Mil’man, V.D., Geometric theory of Banach spaces. Part I: The theory of basis and minimal systems, Russ. Math. Surveys, 1970, vol. 25, no. 3 (153), pp. 113–174.

Zygmund, A., Trigonometric Series, Vol. 2, Cambridge: Cambridge Univ., 1960. Translated under the title Trigonometricheskie ryady, Vol. 2, Moscow: Mir, 1965.

Il’in, V.A. and Joó, I., Inequalities of Bessel and Hausdorff–Young–Riesz type for functions in the class of radial functions, with respect to a system of eigenfunctions of the Laplace operator, Sov. Math., 1986, vol. 34, pp. 474–477.

Karov, Kh.M., Hausdorff–Young inequality for the eigenfunctions and associated functions of a differential operator of second order, Math. Notes, 1987, vol. 42, no. 3, pp. 726–732.

Evzhanov, A.E., Bessel, Hausdorff–Young–Riesz, and Hilbert inequalities for eigenfunctions and associated functions of a second order elliptic operator, Differ. Equations, 1987, vol. 23, no. 10, pp. 1199–1207.

Kurbanov, V.M., Hausdorff–Young inequalities for systems of root vector functions of an nth-order differential operator, Differ. Equations, 1997, vol. 33, no. 3, pp. 355–365.