Prolate spheroidal wave functions of order zero : mathematical tools for bandlimited approximation
 Responsibility
 Andrei Osipov, Vladimir Rokhlin, Hong Xiao.
 Language
 English.
 Publication
 New York : Springer, 2013.
 Physical description
 xi, 379 pages : illustrations ; 25 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 187.
Access
Creators/Contributors
 Author/Creator
 Osipov, Andrei, author.
 Contributor
 Rokhlin, Vladimir, author.
 Xiao, Hong author.
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Introduction
 Mathematical and numerical preliminaries
 Chebyshev systems
 Generalized gaussian quadratures
 Convolutional volterra equations
 Prolate spheroidal wave functions
 The dual nature of PSWFs
 Legendre polyiiomials and PSWFs
 Her mite polynomials and hermite functions
 Recurrence relations
 Hermite functions
 Perturbation of linear operators
 Elliptic integrals
 Oscillation properties of secondorder ODEs
 Growth properties of secondorder ODEs
 Prüfer transformations
 Numerical tools
 Newton's method
 The taylor series method for the solution of ODEs
 A secondorder runge kutta method
 Shifted inverse power method
 Sturm bisection
 Miscellaneous tools
 Overview
 Relation between c, n, and Xn(c)
 Basic facts
 Sharper inequalities involving Xn
 The difference Xm(c)Xn(c)
 Approximate formulas for Xn(c)
 Relation between c, n, and ... n(c)
 Basic facts
 Explicit upper bounds on ...
 Approximate formulas for ... n(c)
 Additional properties of ... (c)
 Properties of PSWFs
 Basic facts
 Oscillation properties of PSWFs
 Growth properties of PSWFs
 Approximate formulas for PSWFs
 PSWFs and the fourier transform
 PSWFs and the bandlimited functions
 PSWFbased quadrature rules
 Generalized gaussian quadrature rules
 Quadrature rules based on the euclidean algorithm
 Quadrature rules based on partial fraction expansion
 Comparison of various PSWFbased quadrature rules
 Additional properties of PSWFbased quadrature rules
 Analysis of a differential operator
 Summary
 Oscillation properties of PSWFs
 Special points of ...n
 A sharper inequality for Xn
 A certain transformation of a prolate ODE
 Further improvements
 Growth properties of PSWFs
 Numerical results
 Analysis of the integral operator
 Summary and discussion
 Summary of analysis
 Accuracy of upper bounds on ...
 Analytical tools
 Legendre expansion
 Principal result : an upper bonnd on ...
 Weaker but simpler bounds
 Numerical results
 Rational approximations of PSWFs
 Overview of the analysis
 Oscillation properties of PSWFs outside (1,1)
 Growth properties of PSWFs outside (1,1)
 Transformation of a prolate ODE into a 2 x 2 system
 The behavior of ...n in the upper halfplane
 Partial fraction expansion of ...n
 The first few terms of the expansion
 The tail of the expansion
 The cauchy boundary term
 Numerical results
 Illustration of results from sect. 6.2
 Illustration of results from sect. 6.3
 Illustration of results from sect. 6.4
 Miscellaneous properties of PSWFs
 The ratio ...m ...n
 Decay of legendre coefficients of PSWFs
 Additional properties
 Asymptotic analysis of PSWFs
 Introduction
 Analytical tools
 Inverse power method as an analytical tool
 Connections between ...m(1) and ...m for large m
 Formulas based on legendre series
 Conclusions
 Formulas based on WKB analysis of the prolate ODE
 Formulas based on hermite series
 Introduction
 Expansion of PSWFs into a hermite series
 Asymptotic expansions for prolate functions
 Asymptotic expansions for eigenvalues Xm
 Error estimates
 Conclusions
 Numerical results
 Numerical results related to sects. 8.3 and 8.4
 Numerical results related to sect. 8.5
 quadrature rules and Interpolation via PSWFs
 Generalized gaussian quadrature rules
 Quadrature rules based on the euclidean algorithm
 Euclidean algorithm for bandlimited functions
 Quadrature nodes from the division theorem
 Interpolation via PSWFs
 Outline
 Intuition behind quadrature weights
 Overview of the analysis
 Analytical tools
 Expansion of ... into a prolate series
 Quadrature error
 The principal result
 Quadrature weights
 Miscellaneous properties of Quadrature weights
 Numerical results
 Illustration of results from sects. 9.19.3
 Illustration of results from sect. 9.4.4
 Quadrature error and its relation to ...
 Quadrature weights
 Generalizations and conclusions
 Numerical algorithms
 Simultaneous evaluation of Xm, ...m, ...m for multiple m
 Evaluation of ... for multiple m
 Evaluation of ... for ... given ...
 Simultaneous evaluation of ...m for multiple m
 Evaluation of Xn and ...n(x), ...n(x) for ... and a single n
 Evaluation of Xn and ... for a single n
 (Initial approximation ...n of ...n)
 (Evaluation of Xn and ...)
 Evaluation of ... for ... given ..
 Evaluation of ...n for a single n
 Evaluation of the quadrature nodes from sect. 9.4
 Evaluation of the quadrature weights from sect. 9.4
 Evaluation of ... and Its roots outside (1,1)
 Evaluation of ... for ...
 Evaluation of ... for ...
 Evaluation of the roots of ...
 Bibliography
 Index.
 Publisher's 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, selfcontained resource for both theory and computation. It will be of interest to a wide range of scientists and engineers, from mathematicians interested in PSWFs as an analytical tool to electrical engineers designing filters and antennas.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Title Variation
 Mathematical tools for bandlimited approximation
 Series
 Applied mathematical sciences, 00665452 ; v. 187
 Available in another form
 9781461482598 (online)
 (GyWOH)har135025973
 ISBN
 9781461482581 (hd.bd.)
 9781461482598 (online)
 1461482585 (hd.bd.)
 1461482593