## Approximation Theory and Approximation Practice

This textbook, with 163 figures and 210 exercises, was published in 2013. It is available from SIAM and from Amazon.

Unusual features:

• The emphasis is on topics close to numerical algorithms.
• Everything is illustrated with Chebfun.
• Each chapter is a PUBLISHable Matlab M-file.
• There is a bias toward theorems and methods for analytic functions, which appear so often in practical approximation, rather than on functions at the edge of discontinuity with their seductive theoretical challenges.
• Original sources are cited rather than textbooks, and each item in the 27-page bibliography is annotated with an editorial comment.
The first six chapters are available online.

All the M-files used to generate the book with Matlab PUBLISH are also available. This makes it easy to run any of the numerical demonstrations from the book, assuming you have Chebfun in your Matlab path. You can get them from a zip file or individually:

• chap1.m    Introduction
• chap2.m    Chebyshev points and interpolants
• chap3.m    Chebyshev polynomials and series
• chap4.m    Interpolants, truncations, and aliasing
• chap5.m    Barycentric interpolation formula
• chap6.m    Weierstrass approximation theorem
• chap7.m    Convergence for differentiable functions
• chap8.m    Convergence for analytic functions
• chap9.m    Gibbs phenomenon
• chap10.m  Best approximation
• chap11.m  Hermite integral formula
• chap12.m  Potential theory and approximation
• chap13.m  Equispaced points, Runge phenomenon
• chap14.m  Discussion of higher-order polynomial interpolation
• chap15.m  Lebesgue constants
• chap16.m  Best and near-best
• chap17.m  Orthogonal polynomials
• chap18.m  Polynomial roots and colleague matrices
• chap19.m  Clenshaw-Curtis and Gauss quadrature
• chap20.m  Carathéodory-Fejér approximation
• chap21.m  Spectral methods
• chap22.m  Linear approximations: beyond polynomials
• chap23.m  Nonlinear approximations: why rational functions?
• chap24.m  Rational best approximation
• chap25.m  Two famous problems
• chap26.m  Rational interpolation and linearized least-squares
• chap27.m  Padé approximation
• chap28.m  Analytic continuation and convergence acceleration
• refs.m       References
ERRATA
p 11: Exercise 2.2: in the final formula N should be n
p 22: Exercise 3.6: the exponent k-1 should be (k-1)/2
p 26: subscripts m should be n; -k(mod 2n) should be (-k)(mod 2n)
p 30: Exercise 4.4(d): length(f(np)) should be length(f(Mmax+1))
p 31: Exercise 4.6 should insert "(down to machine precision, in practice, by Chebyshev interpolation)" before "and then"
p 47: Exercise 6.6(b): 2n should be 2n-1 (6 times)
p 51: 1.652783663415789e+04 should be 2.102783663403057e+04
p 54: Exercise 7.6(b): s=linspace(-1,1,10), p=chebfun(@(x) spline(s,exp(s),x));
p 57: just before the second displayed equation, (3.12) should be (3.13)
p 71: Exercise 9.8: sign(sin(x/2)) should be sign(x)
p 74: Bolzano-Weierstrass should be Heine-Borel
p 78: Exercise 10.1: after 'splitting','on' insert ,'minsamples',65
p 82: Cauchy stated a related formula but not exactly "the same result"
p 93: the product in (12.14) should run over j<k, not j≠k
p 119: the pointer to Exercise 10.5 should be to Exercise 10.6
p 127: the formulas need to be normalized by division by terms like (p_k,p_k)
pp 147, 151: Eqs. (19.10), (19.12) are incorrectly copied from Trefethen (2008): (n-2ν-1)^{2ν+1} should be (2n+1-ν)^ν
p 160: "maps [-1,1]" should be "maps the unit circle"
p 166: Eq. (21.2) is incorrect (p. 166 and inside back cover)
p 215: the integral in (25.13) should have limits from -∞ to ∞
p 215: after (25.14), "even number" should be "odd number"
p 215: on the last line, type (n,n) should be type (n-1,n)
p 222: in (26.3), r(z) should be r(x)
p 229: the summation at the bottom needs a square root
p 256: Lottka should be Lotka
p 296: de la Vallée Poussin (1910) is missing an annotation
p 300: Borel should also list page 75
p 304: Richardson extrapolation should list pages 257-258
p 305: Weierstrass should not list page 75

CHANGES NEEDED FOR CHEBFUN VERSION 5
p 168: [0 0 0 1 0][0 0 0 1 0]'

OTHER NOTES
p 151: concerning Xiang and Bornemann [2012], Bornemann has pointed out (personal communication, August 2013) that just the right result along these lines, derived from L1 approximation, appeared years ago as Theorem 2 in G. Freud, "Über einseitige Approximation durch Polynome. I," Act. Sci. Math. Szeged 16 (1955), 12-28.