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Prolate spheroidal wave functions of order zero : mathematical tools for bandlimited approximation / Andrei Osipov, Vladimir Rokhlin, Hong Xiao.

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Author/Creator:
Osipov, Andrei, author.
Language:
English.
Publication date:
2013
Publication:
New York : Springer, 2013.
Format:
  • Book
  • xi, 379 pages : illustrations ; 25 cm.
Title Variation:
Mathematical tools for bandlimited approximation
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 second-order ODEs
  • Growth properties of second-order ODEs
  • Prüfer transformations
  • Numerical tools
  • Newton's method
  • The taylor series method for the solution of ODEs
  • A second-order 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 band-limited functions
  • PSWF-based quadrature rules
  • Generalized gaussian quadrature rules
  • Quadrature rules based on the euclidean algorithm
  • Quadrature rules based on partial fraction expansion
  • Comparison of various PSWF-based quadrature rules
  • Additional properties of PSWF-based 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 half-plane
  • 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 band-limited 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. 9.1-9.3
  • 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.
Contributor:
Rokhlin, Vladimir, author.
Xiao, Hong, author.
Available in another form:
9781461482598 (online)
(GyWOH)har135025973
Series:
Applied mathematical sciences, 0066-5452 ; v. 187
Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 187.
Subjects:
ISBN:
9781461482581
1461482585

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