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QA405 .F55 O85 2013
Solved problems
1963
Farrell, Orin J.
Math & Statistics Library » QA405 .F3
Spheroidal wave functions
1957
Flammer, Carson
Engineering Library (Terman) » QA405 .F55
Prolate spheroidal wave functions o...
2013
Osipov, Andrei
Math & Statistics Library » QA405 .F55 O85 2013
Subharmonic functions
1976
Hayman, W. K. (Walter Kurt), 1926
Math & Statistics Library » QA405 .H39 V.2
Theory and application of Mathieu f...
1947
McLachlan, N. W. (Norman William), 1888
Math & Statistics Library » QA405 .M3
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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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QA405 .F55 O85 2013
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QA405 .F55 O85 2013
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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 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.
Contributor:
Rokhlin, Vladimir,
author.
Xiao, Hong,
author.
Available in another form:
9781461482598 (online)
(GyWOH)har135025973
Series:
Applied mathematical sciences, 00665452 ; v. 187
Applied mathematical sciences (SpringerVerlag New York Inc.) ;
v. 187.
Subjects:
Spheroidal functions.
Wave functions.
Zero (The number)
ISBN:
9781461482581
1461482585
1461482593
Catkey: 10317681
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