Hilbert Space, Boundary Value Problems and Orthogonal Polynomials

1. Front Matter

   Pages i-xiv

   PDF

2. Part 1

   1. Hilbert Spaces

      • Allan M. Krall

      Pages 1-16

   2. Bounded Linear Operators On a Hilbert Space

      • Allan M. Krall

      Pages 17-40

   3. Unbounded Linear Operators On a Hilbert Space

      • Allan M. Krall

      Pages 41-50

3. Part 2

   1. Regular Linear Hamiltonian Systems

      • Allan M. Krall

      Pages 51-72

   2. Atkinson’s Theory for Singular Hamiltonian Systems of Even Dimension

      • Allan M. Krall

      Pages 73-85

   3. The Niessen Approach to Singular Hamiltonian Systems

      • Allan M. Krall

      Pages 87-106

   4. Hinton and Shaw’s Extension of Weyl’s M (λ) Theory to Systems

      • Allan M. Krall

      Pages 107-136

   5. Hinton and Shaw’s Extension with Two Singular Points

      • Allan M. Krall

      Pages 137-157

   6. The M(λ) Surface

      • Allan M. Krall

      Pages 159-165

   7. The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point

      • Allan M. Krall

      Pages 167-187

   8. The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points

      • Allan M. Krall

      Pages 189-206

   9. Distributions

      • Allan M. Krall

      Pages 207-221

4. Part 3

   1. Orthogonal Polynomials

      • Allan M. Krall

      Pages 223-235

   2. Orthogonal Polynomials Satisfying Second Order Differential Equations

      • Allan M. Krall

      Pages 237-260

   3. Orthogonal Polynomials Satisfying Fourth Order Differential Equations

      • Allan M. Krall

      Pages 261-279

   4. Orthogonal Polynomials Satisfying Sixth Order Differential Equations

      • Allan M. Krall

      Pages 281-290

   5. Orthogonal Polynomials Satisfying Higher Order Differential Equations

      • Allan M. Krall

      Pages 291-299

   6. Differential Operators in Sobolev Spaces

      • Allan M. Krall

      Pages 301-325

   7. Examples of Sobolev Differential Operators

      • Allan M. Krall

      Pages 327-337

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.

• Boundary value problem
• Differential equations
• Hilbert space
• differential equation
• orthogonal polynomials
• orthogonal poynomials

• Department of Mathematics, The Pennsylvania State University, University Park, USA

  Allan M. Krall

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