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Sertac Goktas,Nazim B. Kerimov,Emir A. Maris 대한수학회 2017 대한수학회지 Vol.54 No.4
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.
Goktas, Sertac,Kerimov, Nazim B.,Maris, Emir A. Korean Mathematical Society 2017 대한수학회지 Vol.54 No.4
The spectral problem $$-y^{{\prime}{\prime}}+q(x)y={\lambda}y,\;0 < x < 1, \atop y(0)cos{\beta}=y^{\prime}(0)sin{\beta},\;0{\leq}{\beta}<{\pi};\;{\frac{y^{\prime}(1)}{y(1)}}=h({\lambda})$$ is considered, where ${\lambda}$ is a spectral parameter, q(x) is real-valued continuous function on [0, 1] and $$h({\lambda})=a{\lambda}+b-\sum\limits_{k=1}^{N}{\frac{b_k}{{\lambda}-c_k}},$$ with the real coefficients and $a{\geq}0$, $b_k$ > 0, $c_1$ < $c_2$ < ${\cdots}$ < $c_N$, $N{\geq}0$. 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.