# Uniform Convergence of Expansions in Root Functions of a Differential Operator with Integral Boundary Conditions

Ordinary Differential Equations

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## Abstract

for a second-order ordinary differential operator with integral boundary conditions on an interval of the real line, we derive conditions for the uniform convergence of the spectral expansion of a function in a series in the system of eigenfunctions and associated functions of the operator. We obtain estimates of the rate of convergence of the series and the rate of equiconvergence of such an expansion of a function and its expansion in the trigonometric Fourier series. We also study the uniform convergence of the expansion of a function in the biorthogonal system.

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## References

- 1.Il’in, V.A.,
*Spektral’naya teoriya differentsial’nykh operatorov*(Spectral Theory of Differential Operators), Moscow: Nauka, 1991.zbMATHGoogle Scholar - 2.Samarskaya, T.A., On a generalization of Steklov’s decomposability theorem to the case of nonlocal boundary conditions,
*Differ. Uravn.*, 1989, vol. 25, no. 11, pp. 2008–2010.MathSciNetzbMATHGoogle Scholar - 3.Kolmogorov, A.N. and Fomin, S.V.,
*Elementy teorii funktsii i funktsional’nogo analiza*(Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1972.zbMATHGoogle Scholar - 4.Krall, A.M., The development of general differential boundary systems,
*Rocky Mt. J. Math.*, 1975, vol. 5, no. 4, pp. 493–542.MathSciNetzbMATHGoogle Scholar - 5.Lomov, I.S., Integral representations of irregular root functions of loaded second-order differential operators,
*Differ. Equations*, 2016, vol. 52, no. 12, pp. 1563–1574.MathSciNetzbMATHGoogle Scholar - 6.Shkalikov, A.A., The basis problem of the eigenfunctions of ordinary differential operators with integral boundary conditions,
*Moscow Univ. Math. Bull.*, 1982, vol. 37, no. 6, pp. 10–20.zbMATHGoogle Scholar - 7.Naimark, M.A.,
*Lineinye differentsial’nye operatory*(Linear Differential Operators), Moscow: Nauka, 1969.Google Scholar - 8.Khromov, A.P., Eigenfunction expansion of ordinary linear differential operators in a finite interval,
*Sov. Math. Dokl.*, 1962, vol. 3, pp. 1510–1514.Google Scholar - 9.Khromov, A.P., Eigenfunction expansion of ordinary linear differential operators with irregular decomposing boundary conditions,
*Mat. Sb.*, 1966, vol. 70, no. 3, pp. 310–329.MathSciNetGoogle Scholar - 10.Khromov, A.P., Spectral analysis of differential operators on a finite interval,
*Differ. Equations*, 1995, vol. 31, no. 10, pp. 1657–1662.MathSciNetGoogle Scholar - 11.Sadovnichii, V.A.,
*Analiticheskie metody v spektral’noi teorii differentisial’nykh operatorov*(Analytical Methods in Spectral Theory of Differential Operators), Moscow: Mosk. Gos. Univ., 1973.Google Scholar - 12.Vinokurov, V.A. and Sadovnichii, V.A., Uniform equiconvergence of a Fourier series in eigenfunctions of the first boundary value problem and of a Fourier trigonometric series,
*Dokl. Math.*, 2001, vol. 64, no. 2, pp. 248–252.Google Scholar - 13.Lazetic, N.L., On uniform convergence on closed intervals of spectral expansions and their derivatives for functions from \(W_{p}^{1}\),
*Mat. Vesn.*, 2004, vol. 56, pp. 91–104.MathSciNetzbMATHGoogle Scholar - 14.Kurbanov, V.M. and Safarov, R.A, On uniform convergence of orthogonal expansions in eigenfunctions of Sturm-Liouville operator,
*Trans. Nats. Akad. Nauk Azerb.*, 2004, vol. 24, no. 1, pp. 161–167.MathSciNetzbMATHGoogle Scholar - 15.Il’in, V.A. and Moiseev, E.I., A nonlocal boundary value problem for the Sturm-Liouville operator in the differential and difference treatments,
*Sov. Math. Dokl.*, 1986, vol. 34, no. 3, pp. 507–511.MathSciNetzbMATHGoogle Scholar - 16.Samarskaya, T.A., Absolute and uniform convergence of expansions in eigenfunctions and associated functions of a nonlocal boundary-value problem of the first kind,
*Differ. Equations*, 1989, vol. 25, no. 7, pp. 813–817.MathSciNetzbMATHGoogle Scholar - 17.Lomov, I.S., Uniform convergence of biorthogonal series for the Schrödinger operator with multipoint boundary conditions,
*Differ. Equations*, 2002, vol. 38, no. 7, pp. 941–948.MathSciNetzbMATHGoogle Scholar - 18.Kurbanov, V.M. and Ibadov, E.D., On the properties of systems of root functions of a second-order discontinuous operator,
*Dokl. Math.*, 2009, vol. 80, no. 1, pp. 516–520.MathSciNetzbMATHGoogle Scholar - 19.Kornev, V.V. and Khromov, A.P., On absolute convergence of expansions in eigen- and associated functions of differential and integral operators,
*Dokl. Ross. Akad. Nauk*, 2005, vol. 400, no. 3, pp. 304–308.Google Scholar - 20.Kornev, V.V., Sufficient criterion for absolute and uniform convergence in eigenfunctions of differential operators,
*Sb. Nauch. Trudov Mat. Mekh. Saratov*, 2006, no. 8, pp. 56–58.Google Scholar - 21.Bari, N.K.,
*Trigonometricheskie ryady*(Trigonometric Series), Moscow: Gos. Izd. Fiz. Mat. Lit., 1961.Google Scholar - 22.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.MathSciNetzbMATHGoogle Scholar - 23.Mustafin, M.A., On absolute and uniform convergence of series with respect to a system of sines,
*Differ. Uravn.*, 1992, vol. 28, no. 8, pp. 1465–1466.zbMATHGoogle Scholar - 24.Lomov, I.S., Basis property of the root vectors of loaded second-order differential operators in an interval,
*Differ. Equations*, 1991, vol. 27, no. 1, pp. 64–75.MathSciNetzbMATHGoogle Scholar - 25.Lomov, I.S., Estimates of root functions of the adjoint of a second-order differential operator with integral boundary conditions,
*Differ. Equations*, 2018, vol. 54, no. 5, pp. 596–607.MathSciNetzbMATHGoogle Scholar - 26.Lomov, I.S., Convergence of eigenfunction expansions of a differential operator with integral boundary conditions,
*Dokl. Math.*, 2018, vol. 98, no. 1, pp. 386–390.zbMATHGoogle Scholar - 27.Lomov, I.S., Estimates of the convergence and equiconvergence rate of spectral decompositions of ordinary differential operators,
*Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform.*, 2015, vol. 15, no. 4, pp. 405–418.zbMATHGoogle Scholar

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