Hilbert Space, Boundary Value Problems, and Orthogonal Polynomials

by Krall, Allan M.

• ISBN13: 9783764367015
• ISBN10: 3764367016
• Format: Hardcover
• Copyright: 2002-09-01
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface	xiii
Part 1
Hilbert Spaces
Linear Spaces	1	(2)
Hermitian Forms	3	(3)
Hilbert Spaces	6	(3)
Projections	9	(3)
Continuous Linear Functionals	12	(1)
Orthonormal Sets	13	(2)
Isometric Hilbert Spaces	15	(2)
Bounded Linear Operators on a Hilbert Space
Bounded Linear Operators	17	(2)
The Adjoint Operator	19	(2)
Projections	21	(2)
Some Spectral Theorems	23	(5)
Operator Convergence	28	(5)
The Spectral Resolution of a Bounded Self-Adjoint Operator	33	(4)
The Spectral Resolution of Bounded Normal and Unitary Operators	37	(4)
Normal Operators	37	(2)
Unitary Operators	39	(2)
Unbounded Linear Operators on a Hilbert Space
Unbounded Linear Operators	41	(1)
The Graph of an Operator	42	(2)
Symmetric and Self-Adjoint Operators	44	(2)
The Spectral Resolution of an Unbounded Self-Adjoint Operator	46	(5)
Part 2
Regular Linear Hamiltonian Systems
The Representation of Scalar Problems	51	(3)
Dirac Systems	54	(2)
S-Hermitian Systems	56	(1)
Regular Linear Hamiltonian Systems	57	(7)
The Spectral Resolution of a Regular Linear Hamiltonian Operator	64	(6)
Examples	70	(4)
Atkinson's Theory for Singular Hamiltonian Systems of Even Dimension
Singular Hamiltonian Systems	74	(1)
Existence of Solutions in L2A (a,b)	75	(4)
Boundary Conditions	79	(2)
A Preliminary Greens Formula	81	(7)
The Niessen Approach to Singular Hamiltonian Systems
Boundary Values of Hermitian Forms	88	(3)
The Eigenvalues of A (x)	91	(1)
Generalization of the Second Weyl Theorem	92	(2)
Singular Boundary Value Problems	94	(1)
The Green's Function	95	(2)
Self-Adjointness	97	(3)
Modification of the Boundary Conditions	100	(2)
Other Boundary Conditions	102	(1)
The Limit Point Case	102	(1)
The Limit m Case	103	(1)
The Limit Circle Case	104	(2)
Comments Concerning the Spectral Resolution	106	(1)
Hinton and Shaw's Extension of Weyl's M(λ) Theory to Systems
Notations and Definitions	107	(2)
The M(λ) Matrix	109	(2)
M Circles	111	(4)
Square Integrable Solutions	115	(2)
Singular Boundary Conditions	117	(1)
The Differential Operator L	118	(4)
Extension of the Boundary Conditions	122	(3)
The Extended Green's Formula with One Singular Point	125	(6)
Self-Adjoint Boundary Value Problems with Mixed Boundary Conditions	131	(1)
Examples	132	(6)
Hinton and Shaw's Extension with Two Singular Points
M(λ) Functions, Limit Circles, L2 Solutions	138	(3)
The Differential Operator	141	(1)
The Resolvent, The Green's Function	142	(2)
Parameter Independence of the Domain	144	(1)
The Extended Green's Formula with Two Singular Points	145	(3)
Examples	148	(11)
The Jacobi Boundary Value Problem	149	(1)
The Legendre Boundary Value Problem	150	(1)
The Tchebycheff Problem of the First Kind	150	(1)
The Tchebycheff Problem of the Second Kind	151	(1)
The Generalized Laguerre Boundary Value Problem	152	(1)
The Ordinary Laguerre Boundary Value Problem	152	(1)
The Hermite Boundary Value Problem	152	(1)
Bessel Functions	153	(1)
The Legendre Squared Problem	154	(1)
The Laguerre-Type Problem	155	(4)
The M(λ) Surface
The Connection Between the Hinton-Shaw and Niessen Approaches	159	(3)
A Direct Approach to the M(λ) Surface	162	(2)
Examples	164	(3)
The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point
The Specific Problem	167	(1)
The Spectral Expansion	168	(8)
The Converse Problem	176	(5)
The Relation Between M(λ) and P(λ)	181	(1)
The Spectral Resolution	182	(2)
An Example	184	(1)
Subspace Expansions	185	(2)
Remarks	187	(2)
The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points
The Specific Problem	189	(1)
The Spectral Expansion	190	(9)
The Converse Problem	199	(4)
The Relation Between Ma, Mb, and P(λ)	203	(2)
The Spectral Resolution	205	(2)
Distributions
Test Functions with Compact Support, D; Distributions Without Constraint, D'	207	(4)
Limits of Distributions	211	(1)
Test Functions of Rapid Decay, S; Distributions of Slow Growth, S'	212	(1)
Test Functions of Slow Growth, P; Distributions of Rapid Decay, P'	213	(1)
Test Functions Without Constants, E; Distributions of Compact Support, E'	214	(1)
Distributional Differential Equations	215	(8)
Part 3
Orthogonal Polynomials
Basic Properties of Orthogonal Polynomials	223	(3)
Orthogonal Polynomials, Differential Equations, Symmetry Factors and Moments	226	(11)
Orthogonal Polynomials Satisfying Second Order Differential Equations
The General Theory	237	(2)
The Jacobi Polynomials	239	(4)
The Legendre Polynomials	243	(2)
The Generalized Laguerre Polynomials	245	(4)
The Hermite Polynomials	249	(3)
The Generalized Hermite Polynomials	252	(5)
The Generalized Hermite Polynomials of Even Degree	253	(2)
The Generalized Hermite Polynomials of Odd Degree	255	(2)
The Bessel Polynomials	257	(4)
Orthogonal Polynomials Satisfying Fourth Order Differential Equations
The General Theory	261	(1)
The Jacobi Polynomials	262	(2)
The Generalized Laguerre Polynomials	264	(1)
The Hermite Polynomials	265	(1)
The Legendre-Type Polynomials	265	(5)
The Laguerre-Type Polynomials	270	(4)
The Jacobi-Type Polynomials	274	(7)
Orthogonal Polynomials Satisfying Sixth Order Differential Equations
The H. L. Krall Polynomials	281	(6)
The Littlejohn Polynomials	287	(2)
The Second Littlejohn Polynomials	289	(1)
Koekoek's Generalized Jacobi Type Polynomials	290	(1)
Orthogonal Polynomials Satisfying Higher Order Differential Equations
The Generalized Jacobi-Type Polynomials	291	(4)
The Generalized Laguerre-Type Polynomials {Lαn M (x)}∞n=0	295	(1)
The Generalized Laguerre-Type Polynomials {L2n(1/R)_ (x)}∞n=0	296	(6)
Differential Operators in Sobolev Spaces
Regular Second Order Sobolev Boundary Value Problems	302	(5)
Regular Sobolev Boundary Value Problems for Linear Hamiltonian Systems	307	(5)
Singular Second Order Sobolev Boundary Value Problems	312	(15)
Examples of Sobolev Differential Operators
Regular Second Order Operators	327	(1)
Regular Hamiltonian Systems	328	(2)
Singular Second Order Sobolev Boundary Value Problems	330	(9)
The Laplacian Operators	330	(1)
The Bessel Operators	331	(2)
The Jacobi Operator	333	(1)
The Generalized Laguerre Operator	334	(2)
The Hermite Operator	336	(1)
The Generalized Even Hermite Operator	336	(1)
The Generalized Odd Hermite Operator	336	(3)
The Legendre-Type Polynomials and the Laguerre-Type Polynomials in a Sobolev Space
The Legendre-Type Polynomials	339	(1)
The Laguerre-Type Polynomials	340	(1)
Remarks	341	(2)
Closing Remarks	343	(2)
Index	345