Ebook: Hilbert Space, Boundary Value Problems and Orthogonal Polynomials

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.

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This monograph consists of three parts: - the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators; - the application to linear Hamiltonian systems, giving the details of the spectral resolution; - further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.

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This monograph consists of three parts: - the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators; - the application to linear Hamiltonian systems, giving the details of the spectral resolution; - further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.
Content:
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 (?) Theory to Systems....Pages 107-136
Hinton and Shaw’s Extension with Two Singular Points....Pages 137-157
The M(?) 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

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This monograph consists of three parts: - the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators; - the application to linear Hamiltonian systems, giving the details of the spectral resolution; - further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.
Content:
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 (?) Theory to Systems....Pages 107-136
Hinton and Shaw’s Extension with Two Singular Points....Pages 137-157
The M(?) 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
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