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 Mfile.
 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 27page 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 k1 should be (k1)/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 2n1 (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: BolzanoWeierstrass should be HeineBorel
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): (n2ν1)^{2ν+1} should be (2n+1ν)^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 (n1,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 287: Mergelyan (1951), Adak. should be Akad.
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 257258
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 L^{1} approximation, appeared years
ago as Theorem 2 in G. Freud, "Über einseitige Approximation
durch Polynome. I," Act. Sci. Math. Szeged 16 (1955), 1228.
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