On the Uniform Convergence of Fourier Series Expansions for Sturm–Liouville Problems with a Spectral Parameter in the Boundary Conditions

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.…”

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.)…”

A study on the uniform convergence of spectral expansions for continuous functions on a Sturm-Liouville problem

“…Sturm-Liouville operators with the boundary condition (or conditions) depending on the eigenparameter, one of a concrete classes of this linear differential operators, have been studied to research their various properties by many authors. For example, problems of proving existence of eigenvalues, obtaining oscillation of eigenfunctions, giving asymptotic formulae of eigenvalues and eigenfunctions have been studied in papers [3,5]; problems of investigating the basis properties of the system of root functions in L p .0; 1/ ; 1 < p < 1 have been investigated in papers [1,13]; problems of obtaining the convergence conditions of the spectral expansions in terms of root functions in given some linear spaces has been proved in papers [6,9,10,[13][14][15][16] for one boundary value problem.…”