## Approximation Theory and Approximation Practice

This textbook, with 163 figures and 210 exercises, was first published in 2013. The Extended Edition appeared in 2019. It is available from Amazon and from SIAM, where it is part of the eBooks package.

"ATAP" focuses on the Chebyshev case of approximation of nonperiodic functions on an interval. The Extended Edition includes two appendices concerning the parallel situations of Fourier approximation of periodic functions and Laurent approximation of functions on the unit circle. To learn more read the Preface to the Extended Edition. The Extended Edition also corrects errors and updates the Chebfun syntax.

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. For more information go to the Chebfun web pages. Note that video lectures from this book are available online: see Trefethen homepage.

ERRATA
p. 114: [1:10 10:19] should be [1:10 10:19]'
p. 142: 'randn(201,1)' should be randn(201,1) in Ex. 18.7
p. 168: [0 0 0 1 0] should be [0 0 0 1 0]'
p. 185: f(g(s))*gp(s) should be f(g(s)).*gp(s)
p. 226: : should be ':' in the definitions of Ctilde and C

ERRATA IN THE 2013 EDITION (updated at the Chebfun web pages).
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 62: Exercise 8.10: kappa < M/m should be kappa > M/m
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)
p 138: (18.1) should be (18.2)
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 222: the residue is (4/9)epsilon, not (4/3) epsilon
p 222: on the last line, the denominator should be 1 + (4/3)epsilon
p 229: the summation at the bottom needs a square root
p 256: Lottka should be Lotka
p 294: Tietze (1917), delete "Angew."
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.

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