Krall A.M. Hilbert Space, Boundary Value Problems and Orthogonal Polynomials

Basel: Birkhäuser, 2002. - 354p.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 in­ 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 fundamen­ tal papers by Professor Everitt. They are held in highest esteem, and are given profound thanks.Hilbert Spaces
Bounded Linear Operators On a Hilbert Space
Unbounded Linear Operators On a Hilbert Space
Regular Linear Hamiltonian Systems
Atkinson’s Theory for Singular Hamiltonian Systems of Even Dimension
The Niessen Approach to Singular Hamiltonian Systems
Hinton and Shaw’s Extension of Weyl’s M (λ) Theory to Systems
Hinton and Shaw’s Extension with Two Singular Points
The M(λ) Surface
The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point
The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points
Distributions
Orthogonal Polynomials
Orthogonal Polynomials Satisfying Second Order Differential Equations
Orthogonal Polynomials Satisfying Fourth Order Differential Equations
Orthogonal Polynomials Satisfying Sixth Order Differential Equations
Orthogonal Polynomials Satisfying Higher Order Differential Equations
Differential Operators in Sobolev Spaces
Examples of Sobolev Differential Operators
The Legendre-Type Polynomials and the Laguerre-Type Polynomials in a Sobolev Space