Approximation Theory and Approximation PracticeThis book presents a twenty-first century approach to classical polynomial and rational approximation theory. The reader will find a strikingly original treatment of the subject, completely unlike any of the existing literature on approximation theory, with a rich set of both computational and theoretical exercises for the classroom. There are many original features that set this book apart: the emphasis is on topics close to numerical algorithms; every idea is illustrated with Chebfun examples; each chapter has an accompanying Matlab file for the reader to download; the text focuses on theorems and methods for analytic functions; original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. This textbook is ideal for advanced undergraduates and graduate students across all of applied mathematics. |
Contents
Introduction | 1 |
Chebyshev Points and Interpolants | 7 |
Chebyshev Polynomials and Series | 13 |
Interpolants Projections and Aliasing | 25 |
Barycentric Interpolation Formula | 33 |
Weierstrass Approximation Theorem | 43 |
Convergence for Differentiable Functions | 49 |
Convergence for Analytic Functions | 55 |
ClenshawCurtis and Gauss Quadrature | 143 |
CarathéodoryFejér Approximation | 155 |
Spectral Methods | 165 |
Beyond Polynomials | 177 |
Why Rational Functions? | 189 |
Rational Best Approximation | 199 |
Two Famous Problems | 209 |
Rational Interpolation and Linearized LeastSquares | 221 |
Gibbs Phenomenon | 63 |
Best Approximation | 73 |
Hermite Integral Formula | 81 |
Potential Theory and Approximation | 89 |
Equispaced Points Runge Phenomenon | 95 |
Discussion of HighOrder Interpolation | 103 |
Lebesgue Constants | 107 |
Best and NearBest | 117 |
Orthogonal Polynomials | 123 |
Polynomial Roots and Colleague Matrices | 133 |
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Approximation Theory and Approximation Practice, Extended Edition Lloyd N. Trefethen Limited preview - 2019 |
Common terms and phrases
accuracy algorithm analysis analytic appear applications approximation theory barycentric Bernstein best approximation bounded called Chapter Chebfun Chebyshev interpolants Chebyshev points Chebyshev polynomials circle Clenshaw–Curtis close coefficients command compute condition consider construct continuous contour convergence corresponding curve defined derivative difference differential discussion effect eigenvalues ellipse equal equations equioscillation equispaced error evaluate exact example Exercise exponential fact formula function f Gauss quadrature given gives grid hold idea integral interval known Lagrange Lebesgue constants Legendre linear look Math mathematics Matlab matrix methods norm Note null vector numerical operator optimal Padé approximation plot poles polynomial interpolation practical precision problem projection proof proved quadrature rational approximation rational functions remez result roots satisfy singularities solution square Suppose Taylor Theorem transformed Trefethen unique unit values vector weights zeros