ON THE UNIFORM CONVERGENCE OF SPECTRAL EXPANSIONS FOR A SPECTRAL PROBLEM WITH A BOUNDARY CONDITION RATIONALLY DEPENDING ON THE EIGENPARAMETER
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초록
The spectral problem \[\begin{matrix} -{y}''+q(x)y=\lambda y,{ }0<x<1, \\ y(0)\cos \beta ={y}'(0)\sin \beta ,{ }0\le \beta <\pi ;{ }\frac{{y}'(1)}{y(1)}=h(\lambda ), \\ \end{matrix}{ }\] is considered, where $\lambda $ is a spectral parameter, $q(x)$ is real-valued continuous function on $[0,1]$ and \[h(\lambda )=a\lambda +b-\sum_{k=1}^{N}{\frac{{{b}_{k}}}{\lambda -{{c}_{k}}}},\] with the real coefficients and $a\ge 0,{{b}_{k}}>0,{{c}_{1}}<{{c}_{2}}<\cdots<{{c}_{N}},N\ge 0.$ The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.
Abstract
The spectral problem \[\begin{matrix} -{y}''+q(x)y=\lambda y,{ }0<x<1, \\ y(0)\cos \beta ={y}'(0)\sin \beta ,{ }0\le \beta <\pi ;{ }\frac{{y}'(1)}{y(1)}=h(\lambda ), \\ \end{matrix}{ }\] is considered, where $\lambda $ is a spectral parameter, $q(x)$ is real-valued continuous function on $[0,1]$ and \[h(\lambda )=a\lambda +b-\sum_{k=1}^{N}{\frac{{{b}_{k}}}{\lambda -{{c}_{k}}}},\] with the real coefficients and $a\ge 0,{{b}_{k}}>0,{{c}_{1}}<{{c}_{2}}<\cdots<{{c}_{N}},N\ge 0.$ The sharpened asymptotic formulae for eigenvalues and eigenfunctions of above-mentioned spectral problem are obtained and the uniform convergence of the spectral expansions of the continuous functions in terms of eigenfunctions are presented.
참고문헌 (17)
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