On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

Abstract

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problemwhere is a spectral parameter, q.

The uniform convergence of Fourier series expansions of a Sturm-Liouville problem with boundary condition which contains the eigenparameter

“…where is an eigenparameter, q(x) 2 L 1 (0; 1) is a complex-valued function, a and b are arbitrary complex numbers which satisfy the condition jaj + jbj 6 = 0. In the spectral theory of di¤erential operators, there are many articles containing Sturm-Liouville equations with boundary conditions linearly or polynomially dependent on the spectral parameter [6,7,3,4,5,1,2,9,10,11,12,17]. The convergence conditions of Fourier series expansions of functions in some functional class of Sturm-Liouville operators are investigated in [6,7,3,4,5,1,2,9,10].…”

“…In the spectral theory of di¤erential operators, there are many articles containing Sturm-Liouville equations with boundary conditions linearly or polynomially dependent on the spectral parameter [6,7,3,4,5,1,2,9,10,11,12,17]. The convergence conditions of Fourier series expansions of functions in some functional class of Sturm-Liouville operators are investigated in [6,7,3,4,5,1,2,9,10]. For example, the convergence conditions of series expansions of the following problems are studied in [1], [6], [7], respectively: spectral problems that appear modeling heat transfer in a homogeneous rod with a linear relation between the heat ‡ux and temperature at one endpoint and with a lumped heat capacity at the other endpoint, spectral problems that appear in a model of a transrelaxation heat process and in the mathematical description of vibrations of a loaded string and, spectral problems that appear on vibrations of a homogeneous loaded string, torsional vibrations of a rod with a pulley at one end, heat propagation in a rod with lumped heat capacity at one end, and the current in a cable grounded at one end through a concentrated capacitance or inductance.…”

Uniform Convergence of Spectral Expansions in the Terms of Root Functions of a Spectral Problem for the Equation of a Vibrating Beam

“…The spectral properties (including the basis properties of the root functions in the space L p (0,1), 1 < p < ∞) of ordinary differential operators of second and fourth orders with spectral parameter in the boundary conditions have been studied in papers [1][2][3][4][5][8][9][10][11] (see also their references). There are many articles that investigate the uniform convergence of the expansions for the functions in terms of root functions of Sturm-Liouville operators with a spectral parameter in the boundary conditions (see, for example, [8,10,12,13]). The uniform convergence of Fourier series expansions in the terms of root functions for the differential operators of fourth order with spectral parameter in the boundary conditions apparently, so far, have not been investigated.…”

On the spectral properties of a Sturm-Liouville problem with eigenparameter in the boundary condition

“…Sturm-Liouville problems with the boundary conditions depending on the spectral parameter were studied in order to investigate their various properties in many articles (see [1,2,[6][7][8][10][11][12][13][14][15][16][17][19][20][21][22][23]25]). The problems on the basis property of system of root functions corresponding to Sturm-Liouville problems for some differential operators which contain different forms (linearly, rationally, quadratically etc.)…”