Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation

385 Pages·2013·2.51 MB·English

by  Andrei Osipov

Applied Mathematical Sciences
Andrei Osipov
Vladimir Rokhlin
Hong Xiao
Prolate
Spheroidal
Wave Functions
of Order Zero
Mathematical Tools for Bandlimited
Approximation
Applied Mathematical Sciences
Volume 187
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Forfurthervolumes:
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Andrei Osipov • Vladimir Rokhlin • Hong Xiao
Prolate Spheroidal Wave
Functions of Order Zero
Mathematical Tools for Bandlimited
Approximation
123
AndreiOsipov VladimirRokhlin
DepartmentofMathematics DepartmentofComputerScience
YaleUniversity YaleUniversity
NewHaven,CT,USA NewHaven,CT,USA
HongXiao
DepartmentofComputerScience
UniversityofCalifornia
Davis, CA,USA
ISSN0066-5452
ISBN978-1-4614-8258-1 ISBN978-1-4614-8259-8(eBook)
DOI10.1007/978-1-4614-8259-8
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Preface
Historically,specialfunctionshaveplayedacentralroleinclassicalanalysisand
mathematicalphysics. Theseusuallyarisewhenthepartialdifferentialequation
describingacertainphysicalphenomenonissolvedviaseparationofvariablesin
the appropriategeometry. Inter alia, special functions providea tool for under-
standingthequalitativebehaviorofsolutionsofthecorrespondingequations;as
such, they have been a subject of extensive study. Famous examples of special
functions include trigonometric and hyperbolic functions, Bessel and Hankel
functions, Airy functions, Mathieu functions, various orthogonal polynomials,
spherical harmonics—to mention just a few.
Prolate spheroidal wave functions are yet another class of special functions.
They are encountered in solving the Helmholtz equation in prolate spheroidal
coordinates via separation of variables. In particular, they are the eigenfunc-
tions of the differential operator defined in (1.1) below. However, they are also
the eigenfunctions of the truncated Fourier transform (see (1.2) below), and as
such,constituteanaturalbasisinthespaceofband-limitedfunctions(i.e.,func-
tionswhoseFouriertransformiscompactlysupported). Inthiscapacity,prolate
spheroidal wave functions naturally occur in signal processing, fluid dynamics,
etc. In other words, they are the eigenfunctions of both a differential opera-
tor and a seemingly unrelated integral operator, corresponding to two distinct
classes of physical phenomena.
Thisremarkablepropertyclearlydistinguishesprolatespheroidalwavefunc-
tions from other classes of special functions. Despite this fact, since the ap-
pearance of the justly famous sequence of papers by Slepian et al. in the 1960s,
prolate spheroidal wave functions have receivedrelatively little attention (com-
pared to other classes of special functions). This seems to be related to the
fact that the classical (“Bouwkamp”) algorithm for their evaluation encounters
numericaldifficultiesinpracticalcomputations. Moreover,anattempttodiago-
nalize the integral operator (1.2) numerically via straightforwarddiscretization
meets with numerical difficulties as well. More specifically, the first several
eigenvalues have essentially the same magnitude, and most of the remaining
ones are below machine accuracy (see Sect.2.4). Thus only a small number of
eigenvectors in the narrow “transition region” can be accurately computed via
such procedures. These flaws overshadow the fact that this approach is also
computationally expensive.
V
VI PREFACE
While the need for a somewhat nontrivial numerical treatment might have
crippled the use of prolate spheroidal wave functions in applications and, to
some degree, contributed to their slightly mysterious reputation, there is prob-
ably an additional reason why they have been inadequately studied. Many of
thepropertiesofprolatespheroidalwavefunctionsarebestunderstoodthrough
a simultaneous analysis of both the differential operator (1.1) and the integral
operator (1.2); moreover, it is the interplay between these two types of anal-
ysis that leads to nontrivial numerical algorithms. While Slepian et al. took
exactly that approach, much of the subsequent research tends to treat the op-
erators (1.1) and (1.2) in isolation from each other.
Over the last 20 years, an extensive study of prolate spheroidal wave func-
tions via a combination of theoretical analysis and design of numerical algo-
rithms has led to a number of developments. However, this combination has
had an inevitable side effect: at some point, we realized that our papers on the
subjectwerebecomingtoointerconnectedtobepublishedseparately. Whenone
discovers that one’s own papers almost entirely consist of preliminaries, back-
grounds, and connections to previous papers, it is time to write a book. We
present the natural consequence of this observation for the reader’s judgment.
This book is meant to be in the spirit of classical texts such as A Treatise
on the Theory of Bessel Functions, by G.N. Watson, and The Theory of Spher-
ical and Ellipsoidal Harmonics, by E.W. Hobson. Thus, we have touched only
briefly on the wide-ranging applications of prolate spheroidal wave functions,
and insteadconcentratedon their theoretical andcomputational aspects. Also,
we have restricted ourselves to the one-dimensional case. While many of the
resultsinthisbookgeneralizetothe multidimensionalcaseinastraightforward
manner,theanalysisissomewhatlongandinvolved,andisasubjectofongoing
research. It will be published, separately, at a later date.
Acknowledgments
The authors would like to thank Professor Leslie Greengard for his encourage-
ment and support. Also, the authors would like to thank Professors Raphy
Coifman, Peter W. Jones, Yoel Shkolnisky, Amit Singer, and Mark Tygert for
numerous useful discussions.
New Haven, CT Andrei Osipov
New Haven, CT Vladimir Rokhlin
Davis, CA Hong Xiao
Contents
Preface V
1 Introduction 1
2 Mathematical and Numerical Preliminaries 5
2.1 Chebyshev Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Generalized Gaussian Quadratures . . . . . . . . . . . . . . . . . 6
2.3 Convolutional Volterra Equations . . . . . . . . . . . . . . . . . . 7
2.4 Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . . 7
2.5 The Dual Nature of PSWFs . . . . . . . . . . . . . . . . . . . . . 10
2.6 Legendre Polynomials and PSWFs . . . . . . . . . . . . . . . . . 13
2.7 Hermite Polynomials and Hermite Functions . . . . . . . . . . . 18
2.7.1 Recurrence Relations . . . . . . . . . . . . . . . . . . . . 19
2.7.2 Hermite Functions . . . . . . . . . . . . . . . . . . . . . . 20
2.8 Perturbation of Linear Operators . . . . . . . . . . . . . . . . . 21
2.9 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.10 Oscillation Properties of Second-Order ODEs . . . . . . . . . . . 23
2.11 Growth Properties of Second-Order ODEs . . . . . . . . . . . . . 25
2.12 Pru¨fer Transformations . . . . . . . . . . . . . . . . . . . . . . . 26
2.13 Numerical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.13.1 Newton’s Method. . . . . . . . . . . . . . . . . . . . . . . 28
2.13.2 The Taylor Series Method for the Solution of ODEs . . . 28
2.13.3 A Second-Order Runge–Kutta Method . . . . . . . . . . . 28
2.13.4 Shifted Inverse Power Method. . . . . . . . . . . . . . . . 29
2.13.5 Sturm Bisection . . . . . . . . . . . . . . . . . . . . . . . 29
2.14 Miscellaneous Tools . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Overview 33
3.1 Relation Between c, n, and χn(c) . . . . . . . . . . . . . . . . . . 33
3.1.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Sharper Inequalities Involving χn . . . . . . . . . . . . . . 34
3.1.3 The Difference χm(c)−χn(c) . . . . . . . . . . . . . . . . 37
3.1.4 Approximate Formulas for χn(c) . . . . . . . . . . . . . . 40
VII
VIII CONTENTS
3.2 Relation Between c, n, and λn(c) . . . . . . . . . . . . . . . . . . 41
3.2.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Explicit Upper Bounds on |λn(c)| . . . . . . . . . . . . . . 43
3.2.3 Approximate Formulas for λn(c) . . . . . . . . . . . . . . 46
3.2.4 Additional Properties of λn(c) . . . . . . . . . . . . . . . 49
3.3 Properties of PSWFs . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Oscillation Properties of PSWFs . . . . . . . . . . . . . . 52
3.3.3 Growth Properties of PSWFs . . . . . . . . . . . . . . . . 56
3.3.4 Approximate Formulas for PSWFs . . . . . . . . . . . . . 58
3.3.5 PSWFs and the Fourier Transform . . . . . . . . . . . . . 62
3.3.6 PSWFs and the Band-limited Functions . . . . . . . . . . 64
3.4 PSWF-Based Quadrature Rules . . . . . . . . . . . . . . . . . . . 66
3.4.1 Generalized Gaussian Quadrature Rules . . . . . . . . . . 67
3.4.2 Quadrature Rules Based on the Euclidean
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.3 Quadrature Rules Based on Partial Fraction
Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.4 Comparison of Various PSWF-Based
Quadrature Rules . . . . . . . . . . . . . . . . . . . . . . 69
3.4.5 Additional Properties of PSWF-Based
Quadrature Rules . . . . . . . . . . . . . . . . . . . . . . 71
4 Analysis of a Differential Operator 73
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Oscillation Properties of PSWFs . . . . . . . . . . . . . . . . . . 78
4.2.1 Special Points of ψ . . . . . . . . . . . . . . . . . . . . . 78
n
4.2.2 A Sharper Inequality for χ . . . . . . . . . . . . . . . . . 80
n
4.2.3 A Certain Transformation of a Prolate ODE . . . . . . . 95
4.2.4 Further Improvements . . . . . . . . . . . . . . . . . . . . 106
4.3 Growth Properties of PSWFs . . . . . . . . . . . . . . . . . . . . 119
4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 Analysis of the Integral Operator 135
5.1 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . 135
5.1.1 Summary of Analysis . . . . . . . . . . . . . . . . . . . . 135
5.1.2 Accuracy of Upper Bounds on |λn| . . . . . . . . . . . . . 137
5.2 Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2.1 Legendre Expansion . . . . . . . . . . . . . . . . . . . . . 139
5.2.2 Principal Result: An Upper Bound on |λn| . . . . . . . . 151
5.2.3 Weaker but Simpler Bounds . . . . . . . . . . . . . . . . . 156
5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.3.1 Experiment 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . 162
5.3.2 Experiment 5.3.2 . . . . . . . . . . . . . . . . . . . . . . . 164
5.3.3 Experiment 5.3.3 . . . . . . . . . . . . . . . . . . . . . . . 168
CONTENTS IX
6 Rational Approximations of PSWFs 171
6.1 Overview of the Analysis. . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Oscillation Properties of PSWFs Outside (−1,1) . . . . . . . . . 174
6.3 Growth Properties of PSWFs Outside (−1,1) . . . . . . . . . . . 179
6.3.1 Transformation of a Prolate ODE into a 2×2 System . . 179
6.3.2 The Behavior of ψn in the Upper Half-Plane . . . . . . . 182
6.4 Partial Fraction Expansion of 1/ψn . . . . . . . . . . . . . . . . . 190
6.4.1 The First Few Terms of the Expansion . . . . . . . . . . . 190
6.4.2 The Tail of the Expansion . . . . . . . . . . . . . . . . . . 193
6.4.3 The Cauchy Boundary Term . . . . . . . . . . . . . . . . 201
6.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.5.1 Illustration of Results from Sect. 6.2 . . . . . . . . . . . . 208
6.5.1.1 Experiment 6.5.1.1 . . . . . . . . . . . . . . . . . 208
6.5.1.2 Experiment 6.5.1.2 . . . . . . . . . . . . . . . . . 209
6.5.1.3 Experiment 6.5.1.3 . . . . . . . . . . . . . . . . . 210
6.5.2 Illustration of Results from Sect. 6.3 . . . . . . . . . . . . 213
6.5.2.1 Experiment 6.5.2.1 . . . . . . . . . . . . . . . . . 214
6.5.2.2 Experiment 6.5.2.2 . . . . . . . . . . . . . . . . . 214
6.5.2.3 Experiment 6.5.2.3 . . . . . . . . . . . . . . . . . 214
6.5.3 Illustration of Results from Sect. 6.4 . . . . . . . . . . . . 219
6.5.3.1 Experiment 6.5.3.1 . . . . . . . . . . . . . . . . . 219
6.5.3.2 Experiment 6.5.3.2 . . . . . . . . . . . . . . . . . 220
7 Miscellaneous Properties of PSWFs 225
7.1 The Ratio λm/λn . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.2 Decay of Legendre Coefficients of PSWFs . . . . . . . . . . . . . 227
7.3 Additional Properties . . . . . . . . . . . . . . . . . . . . . . . . 230
8 Asymptotic Analysis of PSWFs 243
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2 Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.2.1 Inverse Power Method as an Analytical Tool . . . . . . . 244
8.2.2 Connections Between ψm(1) and λm for Large m . . . . . 246
8.3 Formulas Based on Legendre Series . . . . . . . . . . . . . . . . . 248
8.3.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.4 Formulas Based on WKB Analysis
of the Prolate ODE . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.5 Formulas Based on Hermite Series . . . . . . . . . . . . . . . . . 255
8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.5.2 Expansion of PSWFs into a Hermite Series . . . . . . . . 257
8.5.3 Asymptotic Expansions for Prolate Functions . . . . . . . 260
8.5.4 Asymptotic Expansions for Eigenvalues χm . . . . . . . . 263
8.5.5 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . 264
8.5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 266