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AuthorKrall, Allan M. author
TitleHilbert Space, Boundary Value Problems and Orthogonal Polynomials [electronic resource] / by Allan M. Krall
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2002
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Descript XIV, 354 p. online resource


The following tract is divided into three parts: Hilbert spaces and their (bounded and unbounded) self-adjoint operators, linear Hamiltonian systemsand their scalar counterparts and their application to orthogonal polynomials. In a sense, this is an updating of E. C. Titchmarsh's classic Eigenfunction Expansions. My interest in these areas began in 1960-61, when, as a graduate student, I was introduced by my advisors E. J. McShane and Marvin Rosenblum to the ideas of Hilbert space. The next year I was given a problem by Marvin Rosenblum that involved a differential operator with an "integral" boundary condition. That same year I attended a class given by the Physics Department in which the lecturer discussed the theory of Schwarz distributions and Titchmarsh's theory of singular Sturm-Liouville boundary value problems. I think a Professor Smith was the inยญ structor, but memory fails. Nonetheless, I am deeply indebted to him, because, as we shall see, these topics are fundamental to what follows. I am also deeply indebted to others. First F. V. Atkinson stands as a giant in the field. W. N. Everitt does likewise. These two were very encouraging to me during my younger (and later) years. They did things "right." It was a revelation to read the book and papers by Professor Atkinson and the many fine fundamenยญ tal papers by Professor Everitt. They are held in highest esteem, and are given profound thanks


1 -- I Hilbert Spaces -- II Bounded Linear Operators on a Hilbert Space -- III Unbounded Linear Operators on a Hilbert Space -- 2 -- IV Regular Linear Hamiltonian Systems -- V Atkinsonโ{128}{153}s Theory for Singular Hamiltonian Systems of Even Dimension -- VI The Niessen Approach to Singular Hamiltonian Systems -- VII Hinton and Shawโ{128}{153}s Extension of Weylโ{128}{153}s M(?) Theory to Systems -- VIII Hinton and Shawโ{128}{153}s Extension with Two Singular Points -- IX The M (?) Surface -- X The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point -- XI The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points -- XII Distributions -- 3 -- XIII Orthogonal Polynomials -- XIV Orthogonal Polynomials Satisfying Second Order Differential Equations -- XV Orthogonal Polynomials Satisfying Fourth Order Differential Equations -- XVI Orthogonal Polynomials Satisfying Sixth Order Differential Equations -- XVII Orthogonal Polynomials Satisfying Higher Order Differential Equations -- XVIII Differential Operators in Sobolev Spaces -- XIX Examples of Sobolev Differential Operators -- XX The Legendre-Type Polynomials and the Laguerre-Type Polynomials in a Sobolev Spaces -- Closing Remarks

Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis


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