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.
Author(s): Allan M. Krall (auth.)
Series: Operator Theory: Advances and Applications 133
Edition: 1
Publisher: Birkhäuser Basel
Year: 2002
Language: English
Pages: 354
Tags: Mathematics, general
Front Matter....Pages i-xiv
Hilbert Spaces....Pages 1-16
Bounded Linear Operators On a Hilbert Space....Pages 17-40
Unbounded Linear Operators On a Hilbert Space....Pages 41-50
Regular Linear Hamiltonian Systems....Pages 51-72
Atkinson’s Theory for Singular Hamiltonian Systems of Even Dimension....Pages 73-85
The Niessen Approach to Singular Hamiltonian Systems....Pages 87-106
Hinton and Shaw’s Extension of Weyl’s M (λ) Theory to Systems....Pages 107-136
Hinton and Shaw’s Extension with Two Singular Points....Pages 137-157
The M(λ) Surface....Pages 159-165
The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point....Pages 167-187
The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points....Pages 189-206
Distributions....Pages 207-221
Orthogonal Polynomials....Pages 223-235
Orthogonal Polynomials Satisfying Second Order Differential Equations....Pages 237-260
Orthogonal Polynomials Satisfying Fourth Order Differential Equations....Pages 261-279
Orthogonal Polynomials Satisfying Sixth Order Differential Equations....Pages 281-290
Orthogonal Polynomials Satisfying Higher Order Differential Equations....Pages 291-299
Differential Operators in Sobolev Spaces....Pages 301-325
Examples of Sobolev Differential Operators....Pages 327-337
The Legendre-Type Polynomials and the Laguerre-Type Polynomials in a Sobolev Space....Pages 339-342
Back Matter....Pages 343-352
