MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES

Oktay Sh. Mukhtarov; Kadriye Aydemir

doi:10.11948/2018.1511

2018 Volume 8 Issue 5

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Oktay Sh. Mukhtarov, Kadriye Aydemir. MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1511-1523. doi: 10.11948/2018.1511

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Oktay Sh. Mukhtarov, Kadriye Aydemir. MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1511-1523. doi: 10.11948/2018.1511

MINIMIZATION PRINCIPLE AND GENERALIZED FOURIER SERIES FOR DISCONTINUOUS STURM-LIOUVILLE SYSTEMS IN DIRECT SUM SPACES

1 Department of Mathematics, Faculty of Art and Science, Gaziosmanpaşa University, 60250 Tokat, Turkey;
2 Institute of Mathematics and Mechanics, Azerbaijan National, Academy of Sciences, Baku, Azerbaijan;
3 Department of Mathematics, Faculty of Art and Science, Amasya University, Amasya, Turkey

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Abstract

Abstract

By modifing the Green's function method we study certain spectral aspects of discontinuous Sturm-Liouville problems with interior singularities. Firstly, we define four eigen-solutions and construct the Green's function in terms of them. Based on the Green's function we establish the uniform convergeness of generalized Fourier series as eigenfunction expansion in the direct sum of Lebesgue spaces L₂ where the usual inner product replaced by new inner product. Finally, we extend and generalize such important spectral properties as Parseval equation, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem.
- Sturm-Liouville problems /
- boundary-transmission conditions /
- Rayleigh quotient /
- minimization principle
MSC: 34B24;34L10

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References

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