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E-book
Author Osipov, Andrei, author

Title Prolate spheroidal wave functions of order zero : mathematical tools for bandlimited approximation / Andrei Osipov, Vladimir Rokhlin, Hong Xiao
Published New York : Springer, 2013

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Description 1 online resource (xi, 379 pages) : illustrations
Series Applied Mathematical Sciences, 0066-5452 ; volume 187
Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 187. 0066-5452
Contents 880-01 Mathematical and Numerical Preliminaries -- Overview -- Analysis of the Differential Operator -- Analysis of the Integral Operator -- Rational Approximations of PSWFs.-Miscellaneous Properties of PSWFs --?Asymptotic Analysis of PSWFs -- Quadrature Rules and Interpolation via PSWFs -- Numerical Algorithms
880-01/(S Machine generated contents note: 1. Introduction -- 2. Mathematical and Numerical Preliminaries -- 2.1. Chebyshev Systems -- 2.2. Generalized Gaussian Quadratures -- 2.3. Convolutional Volterra Equations -- 2.4. Prolate Spheroidal Wave Functions -- 2.5. Dual Nature of PSWFs -- 2.6. Legendre Polynomials and PSWFs -- 2.7. Hermite Polynomials and Her mite Functions -- 2.7.1. Recurrence Relations -- 2.7.2. Hermite Functions -- 2.8. Perturbation of Linear Operators -- 2.9. Elliptic Integrals -- 2.10. Oscillation Properties of Second-Order ODEs -- 2.11. Growth Properties of Second-Order ODEs -- 2.12. Prufer Transformations -- 2.13. Numerical Tools -- 2.13.1. Newton's Method -- 2.13.2. Taylor Series Method for the Solution of ODEs -- 2.13.3. Second-Order Runge--Kutta Method -- 2.13.4. Shifted Inverse Power Method -- 2.13.5. Sturm Bisection -- 2.14. Miscellaneous Tools -- 3. Overview -- 3.1. Relation Between c, n, and Χn(c) -- 3.1.1. Basic Facts -- 3.1.2. Sharper Inequalities Involving & Chin -- 3.1.3. Difference Χm(c) -- Χn(c) -- 3.1.4. Approximate Formulas for Χn(c) -- 3.2. Relation Between c, n, and λn(c) -- 3.2.1. Basic Facts -- 3.2.2. Explicit Upper Bounds on λn(c) -- 3.2.3. Approximate Formulas for λn(c) -- 3.2.4. Additional Properties of λ(c) -- 3.3. Properties of PSWFs -- 3.3.1. Basic Facts -- 3.3.2. Oscillation Properties of PSWFs -- 3.3.3. Growth Properties of PSWFs -- 3.3.4. Approximate Formulas for PSWFs -- 3.3.5. PSWFs and the Fourier Transform -- 3.3.6. PSWFs and the Band-limited Functions -- 3.4. PSWF-Based Quadrature Rules -- 3.4.1. Generalized Gaussian Quadrature Rules -- 3.4.2. Quadrature Rules Based on the Euclidean Algorithm -- 3.4.3. Quadrature Rules Based on Partial Fraction Expansion -- 3.4.4. Comparison of Various PSWF-Based Quadrature Rules -- 3.4.5. Additional Properties of PSWF-Based Quadrature Rules -- 4. Analysis of a Differential Operator -- 4.1. Summary -- 4.2. Oscillation Properties of PSWFs -- 4.2.1. Special Points of-ψn -- 4.2.2. Sharper Inequality for Χn -- 4.2.3. Certain Transformation of a Prolate ODE -- 4.2.4. Further Improvements -- 4.3. Growth Properties of PSWFs -- 4.4. Numerical Results -- 5. Analysis of the Integral Operator -- 5.1. Summary and Discussion -- 5.1.1. Summary of Analysis -- 5.1.2. Accuracy of Upper Bounds on λn -- 5.2. Analytical Tools -- 5.2.1. Legendre Expansion -- 5.2.2. Principal Result: An Upper Bound on λn -- 5.2.3. Weaker but Simpler Bounds -- 5.3. Numerical Results -- 5.3.1. Experiment 5.3.1 -- 5.3.2. Experiment 5.3.2 -- 5.3.3. Experiment 5.3.3 -- 6. Rational Approximations of PSWFs -- 6.1. Overview of the Analysis -- 6.2. Oscillation Properties of PSWFs Outside ( ---1,1) -- 6.3. Growth Properties of PSWFs Outside ( ---1,1) -- 6.3.1. Transformation of a Prolate ODE into a 2 [×] 2 System -- 6.3.2. Behavior of ψn in the Upper Half-Plane -- 6.4. Partial Fraction Expansion of 1/ψn -- 6.4.1. First Few Terms of the Expansion -- 6.4.2. Tail of the Expansion -- 6.4.3. Cauchy Boundary Term -- 6.5. Numerical Results -- 6.5.1. Illustration of Results from Sect. 6.2 -- 6.5.1.1. Experiment 6.5.1.1 -- 6.5.1.2. Experiment 6.5.1.2 -- 6.5.1.3. Experiment 6.5.1.3 -- 6.5.2. Illustration of Results from Sect. 6.3 -- 6.5.2.1. Experiment 6.5.2.1 -- 6.5.2.2. Experiment 6.5.2.2 -- 6.5.2.3. Experiment 6.5.2.3 -- 6.5.3. Illustration of Results from Sect. 6.4 -- 6.5.3.1. Experiment 6.5.3.1 -- 6.5.3.2. Experiment 6.5.3.2 -- 7. Miscellaneous Properties of PSWFs -- 7.1. Ratio λm/λn -- 7.2. Decay of Legendre Coefficients of PSWFs -- 7.3. Additional Properties -- 8. Asymptotic Analysis of PSWFs -- 8.1. Introduction -- 8.2. Analytical Tools -- 8.2.1. Inverse Power Method as an Analytical Tool -- 8.2.2. Connections Between ψm(1) and λm for Large m -- 8.3. Formulas Based on Legendre Series -- 8.3.1. Conclusions -- 8.4. Formulas Based on WKB Analysis of the Prolate ODE -- 8.5. Formulas Based on Hermite Series -- 8.5.1. Introduction -- 8.5.2. Expansion of PSWFs into a Hermite Series -- 8.5.3. Asymptotic Expansions for Prolate Functions -- 8.5.4. Asymptotic Expansions for Eigenvalues Χm -- 8.5.5. Error Estimates -- 8.5.6. Conclusions -- 8.6. Numerical Results -- 8.6.1. Numerical Results Related to Sects. 8.3 and 8.4 -- 8.6.2. Numerical Results Related to Sect. 8.5 -- 9. Quadrature Rules and Interpolation via PSWFs -- 9.1. Generalized Gaussian Quadrature Rules -- 9.2. Quadrature Rules Based on the Euclidean Algorithm -- 9.2.1. Euclidean Algorithm for Band-Limited Functions -- 9.2.2. Quadrature Nodes from the Division Theorem -- 9.3. Interpolation via PSWFs -- 9.4. Quadrature Rules Based on Partial Fraction Expansion -- 9.4.1. Outline -- 9.4.2. Intuition Behind Quadrature Weights -- 9.4.3. Overview of the Analysis -- 9.4.4. Analytical Tools -- 9.4.4.1. Expansion of φj into a Prolate Series -- 9.4.4.2. Quadrature Error -- 9.4.4.3. Principal Result -- 9.4.4.4. Quadrature Weights -- 9.5. Miscellaneous Properties of Quadrature Weights -- 9.6. Numerical Results -- 9.6.1. Illustration of Results from Sects. 9.1--9.3 -- 9.6.2. Illustration of Results from Sect. 9.4.4 -- 9.6.2.1. Experiment 9.6.2.1 -- 9.6.2.2. Experiment 9.6.2.2 -- 9.6.3. Quadrature Error and Its Relation to λn -- 9.6.3.1. Experiment 9.6.3.1 -- 9.6.3.2. Experiment 9.6.3.2 -- 9.6.3.3. Experiment 9.6.3.3 -- 9.6.3.4. Experiment 9.6.3.4 -- 9.6.4. Quadrature Weights -- 9.6.4.1. Experiment 9.6.4.1 -- 9.7. Generalizations and Conclusions -- 10. Numerical Algorithms -- 10.1. Simultaneous Evaluation of Χm, ψm, ψm for Multiple m -- 10.1.1. Evaluation of Χm and β0(m), β1(m) ... for Multiple m -- 10.1.2. Evaluation of ψm(x), ψm(x) for -- 1 [≤] x [≤] 1, Given β0(m), β1(m) ... -- 10.2. Simultaneous Evaluation of λm for Multiple m -- 10.3. Evaluation of Χn and ψn(x), ψn(x) for -- 1 [≤] x [≤] 1 and a Single n -- 10.3.1. Evaluation of Χn and β0(n), β1(n) ... for a Single n -- 10.3.1.1. Step 1 (Initial Approximation Χn of Χn) -- 10.3.1.2. Step 2 (Evaluation of Χn and β(n)) -- 10.3.2. Evaluation of ψn(x), ψn(x) for -- 1 [≤] x [≤] 1, Given β0(n), β1(n) ... -- 10.4. Evaluation of λn for a Single n -- 10.5. Evaluation of the Quadrature Nodes from Sect. 9.4 -- 10.6. Evaluation of the Quadrature Weights from Sect. 9.4 -- 10.7. Evaluation of ψn and Its Roots Outside ( --1,1) -- 10.7.1. Evaluation of ψn(x) for x> 1 -- 10.7.2. Evaluation of ψn(x) for x> 1 -- 10.7.3. Evaluation of the Roots of ψn in (1, [∞])
Summary Prolate Spheroidal Wave Functions (PSWFs) are the eigenfunctions of the bandlimited operator in one dimension. As such, they play an important role in signal processing, Fourier analysis, and approximation theory. While historically the numerical evaluation of PSWFs presented serious difficulties, the developments of the last fifteen years or so made them as computationally tractable as any other class of special functions. As a result, PSWFs have been becoming a popular computational tool. The present book serves as a complete, self-contained resource for both theory and computation. It will be of interest to a wide range of scientists and engineers, from mathematicians interested in PSWF as an analytical tool to electrical engineers designing filters and antennas
Bibliography Includes bibliographical references and index
Notes English
Online resource; title from PDF title page (SpringerLink, viewed October 14, 2013)
Subject Spheroidal functions.
Wave functions.
MATHEMATICS -- Calculus.
MATHEMATICS -- Mathematical Analysis.
Spheroidal functions
Wave functions
Form Electronic book
Author Rokhlin, Vladimir, author
Xiao, Hong, author
ISBN 9781461482598
1461482593
1461482585
9781461482581