Geodesic
Basis property of a system of eigenfunctions of a second-order differential operator with involution
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 2, pp. 183-196 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the present paper we study the spectral problem for the second-order differential operators with involution and boundary conditions of Dirichlet type. The Green's function of this boundary problem is constructed. Uniform estimates of the Green's functions for the boundary value problems considered are obtained. The equiconvergence of eigenfunction expansions of two second-order differential operators with involution and boundary conditions of Dirichlet type for any function in $L_{2}(-1,1)$ is established. We use an integral method based on the application of the Green's function of a differential operator with involution and spectral parameter. As a corollary from the equiconvergence theorem, it is proved that the eigenfunctions of the spectral problem form the basis in $L_{2}(-1,1)$ for any continuous complex-valued coefficient $q(x)$.
Keywords: differential equation with involution, Green's function, eigenfunction expansions, basis. 
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A. A. Sarsenbi; B. Kh. Turmetov. Basis property of a system of eigenfunctions of a second-order differential operator with involution. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 2, pp. 183-196. http://geodesic.mathdoc.fr/item/VUU_2019_29_2_a3/

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