Uniform Convergence of Generalized Fourier Series of Hahn-Sturm-Liouville Problem

Authors : Bilender PAŞAOĞLU, Hüseyin TUNA

Pages : 250-259

View : 31 | Download : 7

Publication Date : 2021-10-15

Article Type : Research Paper

Abstract :In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ insert ignore into journalissuearticles values(Ly);\leftinsert ignore into journalissuearticles values( x\right); :=\frac{1}{r\leftinsert ignore into journalissuearticles values( x\right); }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}insert ignore into journalissuearticles values(p\leftinsert ignore into journalissuearticles values( x\right); D_{\omega,q}y\leftinsert ignore into journalissuearticles values( x\right); );+v\leftinsert ignore into journalissuearticles values( x\right); y\leftinsert ignore into journalissuearticles values( x\right); \right] =\lambda y\leftinsert ignore into journalissuearticles values( x\right); ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+insert ignore into journalissuearticles values(a-\omega_{0});q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\leftinsert ignore into journalissuearticles values( \omega_{0}\right); -h_{1}p\leftinsert ignore into journalissuearticles values( \omega_{0}\right); D_{-\omega q^{-1},q^{-1}}y\leftinsert ignore into journalissuearticles values( \omega_{0}\right); =0, y\leftinsert ignore into journalissuearticles values( a\right); +h_{2}p\leftinsert ignore into journalissuearticles values( h^{-1}\leftinsert ignore into journalissuearticles values( a\right); \right); D_{-\omega q^{-1},q^{-1}}y\leftinsert ignore into journalissuearticles values( a\right); =0,$$ where $q\in\leftinsert ignore into journalissuearticles values( 0,1\right); ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}insert ignore into journalissuearticles values(a);}$ and $pinsert ignore into journalissuearticles values(x);>0,$ $r\leftinsert ignore into journalissuearticles values( x\right); >0,\ v\leftinsert ignore into journalissuearticles values( x\right); >0,\ x\in J_{\omega_{0},h^{-1}insert ignore into journalissuearticles values(a);},$ $h^{-1}\leftinsert ignore into journalissuearticles values( a\right); =q^{-1}insert ignore into journalissuearticles values(a-\omega);>a,$ $h^{-1}\leftinsert ignore into journalissuearticles values( \omega _{0}\right); =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+insert ignore into journalissuearticles values(a-\omega _{0});q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.
Keywords : Hahn`s Sturm Liouville equation, Green`s function, Parseval equality, eigenfunction expansion

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