On the convergence in W 2 m of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

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

“…Suppose that the sequence fT m .x/g 1 mDrC1 is the partial sum of the series (56). By using (8), the equality In the case NC1 D d1 c1 , the proof is similar. The third case: Consider the Fourier series f .x/ on the interval OE0, 1 in the system u n .x/ .n D 0, 1, : : : ; n ¤ r/:…”

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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.x/ is a real-valued continuous function on the interval OE0, 1, and a 1 , b 0 , b 1 , c 1 , d 0 , and d 1 are real constants that satisfy the conditionswhere is a spectral parameter, q.x/ is a real-valued continuous function on the interval OE0, 1, and a 1 , b 0 , b 1 , c 1 , d 0 , and d 1 are real constants that satisfy the following conditions:In this article, we study the uniform convergence of the expansions in terms of eigenfunctions of the boundary value problem (1)-(3) for the functions that belong to C OE0, 1. There are many articles that investigate the uniform convergence of the expansions for the functions in terms of root functions of some differential operators with a spectral parameter in the boundary conditions (for example, [1][2][3][4][5][6][7][8]). The condition D a 1 d 1 b 1 c 1 > 0 is essential.

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“…In this article, we study the uniform convergence of the expansions in terms of eigenfunctions of the boundary value problem (1)-(3) for the functions that belong to C OE0, 1. There are many articles that investigate the uniform convergence of the expansions for the functions in terms of root functions of some differential operators with a spectral parameter in the boundary conditions (for example, [1][2][3][4][5][6][7][8]). The condition D a 1 d 1 b 1 c 1 > 0 is essential.…”

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.…”