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Operator Theory: Advances and Applications Vol.133
Editor: I. Gohberg
Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel
Editorial Board: J . Arazy (Haifa) A. Atzmon (Tel Aviv) J . A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P. A. Fillmore (Halifax) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J . A. Helton (La Jolla) M A Kaashoek (Amsterdam) H. G. Kaper (Argonne) ST . Kuroda (Tokyo)
P. Lancaster (Calgary) L E . Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) J . D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J . Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Upmeier (Marburg) S. M. Verduyn Lunel (Amsterdam) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes)
Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Haimos (Santa Clara) T. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)
Hilbert Space, Boundary Value Problems and Orthogonal Polynomials
Allan M. Krall
Springer Basel AG
Author:
Allan M . Krall Department of Mathematics The Pennsylvania State University University Park, PA 16802 USA e-mail: [email protected]
2000 Mathematics Subject Classification 47E05, 33C45, 33C47, 34B05, 37J99
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Krall, Allan M . : Hilbert Space, boundary value problems and orthogonal polynomials / Allan M . Krall. - Basel; Boston ; Berlin : Birkhäuser, 2002
(Operator theory ; Vol. 133) ISBN 978-3-0348-9459-3 ISBN 978-3-0348-8155-5 (eBook) DOI 10.1007/978-3-0348-8155-5
This work is subject to copyright. A l l rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
© 2002 Springer Basel AG Originally published by Birkhäuser Verlag in 2002 Softcover reprint of the hardcover 1st edition 2002 Printed on acid-free paper produced from chlorine-free pulp. TCF «> Cover design: Heinz Hiltbrunner, Basel
9 8 7 6 5 4 3 2 1 www.birkhauser-science.com
To Professor and Mrs. F. V. Atkinson, whose constant encouragement throughout the years
has been a vital source of inspiration.
Contents
Preface
Part!
I Hilbert Spaces
1 Linear Spaces 2 Hermitian Forms .................................................. . 3 Hilbert Spaces ..................................................... .
xiii
1
3 6
4 Projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . 9 5 Continuous Linear Functionals ..................................... 12 6 Orthonormal Sets .................................................. 13 7 Isometric Hilbert Spaces ............................................ 15
II Bounded Linear Operators on a Hilbert Space
1 Bounded Linear Operators ......................................... 17 2 The Adjoint Operator .............................................. 19 3 Projections. . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 21 4 Some Spectral Theorems ........................................... 23 5 Operator Convergence .............................................. 28 6 The Spectral Resolution of a Bounded
Self-Adjoint Operator .............................................. 33 7 The Spectral Resolution of Bounded Normal
and Unitary Operators ............................................. 37 7.1 Normal Operators.. . . . . . . . . . . . .. . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . 37 7.1 Unitary Operators ............................................ 39
III Unbounded Linear Operators on a Hilbert Space
1 Unbounded Linear Operators....................................... 41 2 The Graph of an Operator ......................................... 42 3 Symmetric and Self-Adjoint Operators.............................. 44 4 The Spectral Resolution of an Unbounded
Self-Adjoint Operator .............................................. 46
viii Contents
Part 2
IV Regular Linear Hamiltonian Systems
1 The Representation of Scalar Problems 51 2 Dirac Systems ...................................................... 54 3 S-Hermitian Systems............................................... 56 4 Regular Linear Hamiltonian Systems ............................... 57 5 The Spectral Resolution of a Regular Linear
Hamiltonian Operator .............................................. 64 6 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
V Atkinson's Theory for Singular Hamiltonian Systems of Even Dimension
1 Singular Hamiltonian Systems ...................................... 74 2 Existence of Solutions in L~ (a, b) ................................... 75 3 Boundary Conditions ............................................... 79 4 A Preliminary Greens Formula..................................... 81
VI The Niessen Approach to Singular Hamiltonian Systems
1 Boundary Values of Hermitian Forms ............................... 88 2 The Eigenvalues of A(x) ............................................ 91 3 Generalization of the Second Weyl Theorem ........................ 92 4 Singular Boundary Value Problems ................................. 94 5 The Green's Function .............................................. 95 6 Self-Adjointness. . . . . .. . . .. . . . .. . . . . .. . . . . .. . . . . . . . . . .. .. . . . . .. . . . . . 97 7 Modification of the Boundary Conditions ........................... 100 8 Other Boundary Conditions ........................................ 102 9 The Limit Point Case .............................................. 102 10 The Limit m Case.................................................. 103 11 The Limit Circle Case .............................................. 104 12 Comments Concerning the Spectral Resolution ..................... 106
VII Hinton and Shaw's Extension ofWeyl's M()') Theory to Systems
1 Notations and Definitions .......................................... 107 2 The M ().) Matrix .................................................. 109 3 M Circles .......................................................... 111 4 Square Integrable Solutions ......................................... 115 5 Singular Boundary Conditions ...................................... 117 6 The Differential Operator L ........................................ 118 7 Extension of the Boundary Conditions .............................. 122 8 The Extended Green's Formula with One Singular Point............ 125
Contents ix
9 Self-Adjoint Boundary Value Problems with Mixed Boundary Conditions ........................................ 131
10 Examples .......................................................... 132
VIII Hinton and Shaw's Extension with Two Singular Points
1 M{'\') Functions, Limit Circles, L2 Solutions. . .. . . . . .. . . .. . .. . . .. ... 138
2 The Differential Operator .......................................... 141
3 The Resolvent, The Green's Function............................... 142
4 Parameter Independence of the Domain ............................ 144
5 The Extended Green's Formula with Two Singular Points .......... 145
6 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.1 The Jacobi Boundary Value Problem.. ... . .. ... .... ... . .. . .. .. 149
6.2 The Legendre Boundary Value Problem ....................... 150
6.3 The Tchebycheff Problem of the First Kind ................... 150
6.4 The Tchebycheff Problem of the Second Kind ................. 151
6.5 The Generalized Laguerre Boundary Value Problem........... 152
6.6 The Ordinary Laguerre Boundary Value Problem .............. 152
6.7 The Hermite Boundary Value Problem ........................ 152
6.8 Bessel Functions .............................................. 153
6.9 The Legendre Squared Problem............................... 154
6.10 The Laguerre-Type Problem.................................. 155
IX The M (,\.) Surface
1 The Connection Between the Hinton-Shaw and Niessen Approaches ................................................ 159
2 A Direct Approach to the M{'\') Surface ............................ 162
3 Examples.......................................................... 164
X The Spectral Resolution for Linear Hamiltonian Systems with One Singular Point
1 The Specific Problem ............................................... 167
2 The Spectral Expansion............................................ 168
3 The Converse Problem ............................................. 176
4 The Relation Between M{'\') and P{'\') ............................. 181
5 The Spectral Resolution ............................................ 182
6 An Example ........................................................ 184
7 Subspace Expansions ............................................... 185
8 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
x Contents
XI The Spectral Resolution for Linear Hamiltonian Systems with Two Singular Points
1 The Specific Problem ............................................... 189
2 The Spectral Expansion ............................................ 190
3 The Converse Problem............................................. 199
4 The Relation Between Ma, Mb , and P(A) ........................... 203
5 The Spectral Resolution ............................................ 205
XII Distributions
1 Test Functions with Compact Support, D; Distributions Without Constraint, D' .............................. 207
2 Limits of Distributions ............................................. 211
3 Test Functions of Rapid Decay, S; Distributions of Slow Growth, S' ................................... 212
4 Test Functions of Slow Growth, P; Distributions of Rapid Decay, p' ................................... 213
5 Test Functions Without Constants, E; Distributions of Compact Support, E' .............................. 214
6 Distributional Differential Equations. . . . . . . .. . . . . .. . . . . . . . . . . . . . . . .. 215
Part 3
XIII Orthogonal Polynomials
1 Basic Properties of Orthogonal Polynomials 223
2 Orthogonal Polynomials, Differential Equations, Symmetry Factors and Moments ................................... 226
XIV Orthogonal Polynomials Satisfying Second Order Differential Equations
1 The General Theory ................................................ 237
2 The Jacobi Polynomials............................................ 239
3 The Legendre Polynomials ......................................... 243
4 The Generalized Laguerre Polynomials ............................. 245
5 The Hermite Polynomials .......................................... 249
6 The Generalized Hermite Polynomials .............................. 252
6.1 The Generalized Hermite Polynomials of Even Degree......... 253
6.2 The Generalized Hermite Polynomials of Odd Degree .......... 255
7 The Bessel Polynomials ............................................ 257
Contents
XV Orthogonal Polynomials Satisfying Fourth Order Differential Equations
xi
1 The General Theory................................................ 261 2 The Jacobi Polynomials............................................ 262 3 The Generalized Laguerre Polynomials ............................. 264 4 The Hermite Polynomials .......................................... 265 5 The Legendre-Type Polynomials .................................... 265 6 The Laguerre-Type Polynomials .................................... 270 7 The Jacobi-Type Polynomials...................................... 274
XVI Orthogonal Polynomials Satisfying Sixth Order Differential Equations
1 The H.L. Krall Polynomials........................................ 281 2 The Littlejohn Polynomials ......................................... 287 3 The Second Littlejohn Polynomials ................................. 289 4 Koekoek's Generalized Jacobi Type Polynomials.................... 290
XVII Orthogonal Polynomials Satisfying Higher Order Differential Equations
1 The Generalized Jacobi-Type Polynomials.......................... 291 2 The Generalized Laguerre-Type Polynomials {L~M (x)} ~=o ......... 295
3 The Generalized Laguerre-Type Polynomials {L;(l/R)(x)}~=o 296
XVIII Differential Operators in Sobolev Spaces
1 Regular Second Order Sobolev Boundary Value Problems ........... 302 2 Regular Sobolev Boundary Value Problems
for Linear Hamiltonian Systems .................................... 307 3 Singular Second Order Sobolev Boundary Value Problems.......... 312
XIX Examples of Sobolev Differential Operators
1 Regular Second Order Operators ................................... 327 2 Regular Hamiltonian Systems ...................................... 328 3 Singular Second Order Sobolev Boundary Value Problems .......... 330
3.1 The Laplacian Operators...................................... 330 3.2 The Bessel Operators ......................................... 331 3.3 The Jacobi Operator .......................................... 333 3.4 The Generalized Laguerre Operator ........................... 334 3.5 The Hermite Operator ........................................ 336 3.6 The Generalized Even Hermite Operator...................... 336 3.7 The Generalized Odd Hermite Operator ....................... 336
xii Contents
XX The Legendre-Type Polynomials and the Laguerre-Type Polynomials in a Sobolev Space
1 The Legendre-Type Polynomials .................................... 339 2 The Laguerre-Type Polynomials .................................... 340 3 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 341
Closing Remarks ......................................................... 343
Index ..................................................................... 345
Preface
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 instructor, 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 fundamental papers by Professor Everitt. They are held in highest esteem, and are given profound thanks.
Charles Fulton was responsible for reintroducing the author to the Titchmarsh theory in the late 1970's. His continued support has been invaluable. Angelo Mingarelli, likewise, was also a spur at the right time. Don Hinton and Ken Shaw provided me with the linear Hamiltonian theory, already developed, and it has proved to be invaluable. At the same time, and ever since, Lance Littlejohn and Dick Brown have been marvelous sounding boards, colleagues and friends.
During the mid 1980's, Hans Kaper provided several opportunities at the Argonne National Laboratory. Indeed, he invited me to collaborate with him, Tony Zettl, Derek Atkinson and a number of visitors during the 1986-1987 special year for differential equations at Argonne. It was a tremendous experience. In 1984 I met with Heinz-Dieter Niessen, whose work on differential systems, while slightly different from that of Hinton-Shaw, provided new insight into the nature of boundary value problems.
xiv Preface
I am also indebted to my late father H. L. Krall (1907-1994), who instructed me in the theory of orthogonal polynomials, and, in the late 1970's, mentioned that he had worked out incompletely three sets of orthogonal polynomials and their accompanying 4-th order differential equations. When the author reactivated them, they caused a sensation, especially in Europe. Likewise he mentioned that he had investigated polynomials satisfying a 6-th order differential equation, but had forgotten the details and had never written anything about them. This led to the Lance Littlejohn dissertation. The Europeans took note, and the subject exploded.
Finally the two "little" girls of the late 1980's, Mojdeh Hazmirzaahmad and Alouf Jirari, stimulated me at the right time; they, together with W. N. Everitt and Lance Littlejohn assisted in the development of the "left definite" or Sobolev problems.
I would also like to thank Kathy Wyland for doing a beautiful job typing the manuscript. The mistakes are mine, not hers. The remaining typos, and there must be many, are "deliberately" left so the readers may find them.
Allan M. Krall University Park, PA August 2001
