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

Abstract

In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ 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

References

1. [1] Aldwoah K. A., Generalized time scales and associated di¤erence equa- tions. Ph.D. Thesis, Cairo University (2009).
2. [2] Annaby M. H., Hamza A. E. and Aldwoah K. A., Hahn di¤erence oper- ator and associated Jackson-Nörlund integrals, J. Optim. Theory Appl. 154 (2012), 133–153.
3. [3] Annaby M. A. and Hassan H. A., Sampling theorems for Jackson- Nörlund transforms associated with Hahn-di¤erence operators. J. Math. Anal. Appl. 464 (2018), no. 1, 493–506.
4. [4] Hahn W., Über orthogonalpolynome, die q􀀀Di¤erenzengleichungen genügen, Math. Nachr. 2 (1949), 4–34.
5. [5] Hahn W., Ein Beitrag zur Theorie der Orthogonalpolynome, Monatsh. Math. 95 (1983), 19–24.
6. [6] Hamza A. E. and Ahmed S. A., Theory of linear Hahn di¤erence equa- tions, J. Adv. Math. 4(2) (2013), 440–460.
7. [7] Hamza A. E. and Ahmed S. A., Existence and uniqueness of solutions of Hahn di¤erence equations, Adv. Di¤erence Equations 316 (2013), 1–15.
8. [8] Hamza A. E. and Makharesh S. D., Leibniz’rule and Fubinis theorem associated with Hahn di¤erence operator, Journal of Advanced Mathe- matical, vol. 12, no. 6, (2016), 6335–6345.

9. [9] Annaby M. H., Hamza A. E. and Makharesh S. D., A Sturm-Liouville theory for Hahn di¤erence operator, in: Xin Li, Zuhair Nashed (Eds.), Frontiers of Orthogonal Polynomials and q􀀀Series, World Scienti…c, Singapore, (2018), 35–84.
10. [10] Weyl H., Über gewöhnlicke Di¤erentialgleichungen mit Singuritaten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Annal., 68 (1910), 220-269.
11. [11] Titchmarsh E. C., Eigenfunction Expansions Associated with Second- Order Di¤erential Equations. Part I. Second Edition Clarendon Press, Oxford, 1962.
12. [12] Mukhtarov O. Sh. and AydemirK ., "Basis properties of the eigenfunc- tions of two-interval Sturm–Liouville problems." Analysis and Mathe- matical Physics (2018): 1-20.
13. [13] Aydemir K., Ol¼gar H., Muk htarov O. Sh., Muhtarov F., "Di¤erential operator equations with interface conditions in modi…ed direct sum spaces", Filomat, 32(3), (2018), 921-931.
14. [14] Olgar H., Mukhtarov O. Sh , and Aydemir K., "Some properties of eigenvalues and generalized eigenvectors of one boundary value prob- lem." Filomat 32.3 (2018): 911-920.
15. [15] Aydemir K. and Mukhtarov O. Sh., "Class of Sturm–Liouville Problems with Eigenparameter Dependent Transmission Conditions." Numerical Functional Analysis and Optimization 38.10 (2017): 1260-1275. 17
16. [16] Guseinov G. Sh., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 147–160.
17. [17] Guseinov G. Sh., Eigenfunction expansions for a Sturm-Liouville prob- lem on time scales. Int. J. Di¤erence Equ. 2 (2007), no. 1, 93–104.
18. [18] Faydao¼glu, ¸S. and Guseinov G. Sh., An expansion result for a Sturm- Liouville eigenvalue problem with impulse. Turkish J. Math. 34 (2010), no. 3, 355–366.
19. [19] Huseynov A., Eigenfunction expansion associated with the one- dimensional Schrödinger equation on semi-in…nite time scale intervals. Rep. Math. Phys. 66 (2010), no. 2, 207–235.
20. [20] Allahverdiev B. P. and Tuna H., Spectral expansion for singular Dirac system with impulsive conditions, Turkish J. Math. 42, (2018), no. 5, 2527-2545.
21. [21] Allahverdiev B. P. and Tuna H., An expansion theorem for q􀀀Sturm- Liouville operators on the whole line, Turkish J. Math., 42, (2018), no. 3, 1060-1071.
22. [22] Huseynov A. and Bairamov E., On expansions in eigenfunctions for second order dynamic equations on time scales. Nonlinear Dyn. Syst. Theory 9 (2009), no. 1, 77–88.
23. [23] Naimark M. A., Linear di¤erential operators, 2nd edn, Nauka, Moscow, 1969; English transl. of 1st edn, Parts 1, 2, Ungar, New York, 1967, 1968.
24. [24] Yosida K., On Titchmarsh-Kodaira formula concerningWeyl-Stone