Spectral Theory of Differential Operators Self-Adjoint Differential Operators [Hardcover]

In this fully-illustrated textbook, the author examines the spectral theory of self-adjoint elliptic operators. Chapters focus on the problems of convergence and summability of spectral decompositions about the fundamental functions of elliptic operators of the second order. The author's work offers a novel method for estimation of the remainder term of a spectral function and its Riesz means without recourse to the traditional Carleman technique and Tauberian theorem apparatus.In this fully-illustrated textbook, the author examines the spectral theory of self-adjoint elliptic operators. Chapters focus on the problems of convergence and summability of spectral decompositions about the fundamental functions of elliptic operators of the second order. The author's work offers a novel method for estimation of the remainder term of a spectral function and its Riesz means without recourse to the traditional Carleman technique and Tauberian theorem apparatus.Introduction. Nomenclature. The Expansion in Fundamental System of Functions of the Laplace Operator. The Spectral Expansions Related to an Arbitrary Selfadjoint Nonnegative Extension of the Laplace Operator. On the Riesz Equisummability of Spectral Decompositions in the Classical Sense and Generalized Sense. Selfadjoint Nonnegative Extensions of the Elliptic Operator of the Second Order. Appendix 1: Conditions for Uniform Convergence of Multiple Trigonometric Fourier Series with Spheric Partial Sums. Appendix 2: Conditions for Uniform Convergence of Eigenfunction Expansions of the First, Second, and Third Boundary Problems for the Elliptic Operator of Second Order. Epilogue. Index.