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Minimization principle and Generalized Fourier series for discontinuous Sturm-Liouville systems in direct sum spaces
Keywords:turm-Liouville problems, boundary-transmission conditions, Rayleigh quotient, minimization principle
      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 $L_2$ 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.