N. Akhieser: Theory of approximation. Ungar 1956.

7425 H. Blatt/H. Mashkar: A discrepancy theorem concerning polynomials
of best approximations in L^p_w[-1,1]. Monatsh. 120 (1995), 91-103.

Peter Borwein/Tamas Erdelyi: Polynomials and polynomial inequalities.
Springer 1995, 470p. 3-540-94509-1. DM 98.
"Polynomials and polynomial inequalities is one of the best mathematical
books in years." (Jet Wimp)

Jürgen Braun: On Kolmogorov's superposition theorem and its applications.
SVH 2010, 190p.

27121 Jürgen Braun/Michael Griebel: On a constructive proof of Kolmogorov's
superposition theorem. Constr. Approx. 30/3 (2009), 653-675.

21116 Bruno Brosowski/Frank Deutsch: An elementary proof of the
Stone-Weierstrass theorem. Proc. AMS 81/1 (1981), 89-92.

16249 Martin Buhmann: Radial basis functions. Acta Numerica 9 (2000), 1-38.

E. Cheney: Introduction to approximation theory. Chelsea 1981.

S. Cooper: Continued fractions and orthogonal functions.
400p. $145.

2309 Philip Davis: Interpolation and approximation. Dover 1975.

J.-P. Delahaye: Sequence transformations. Springer 1988, 250p. DM 128.

R. De Vore/G. Lorentz: Constructive approximation.
Springer 1993, 450p. DM 168.

Z. Ditzian/V. Totik: Moduli of smoothness. 
Springer 1987, 220p. DM 108. Carefully written and rather advanced.

5549 Rida Farouki: Computing with barycentric polynomials.
Math. Intell. 13/4 (1991), 61-69.

D. Gaier: Lectures on complex approximation.
Birkhaeuser 1987, 200p. 3-7643-3147-X. SFR 108.

W. Gautschi: Orthogonal polynomials. Oxford UP 2004, 300p. Pds 55.

28383 Armin Iske: Approximation. Springer 2018, 360p. Eur 26.

15009 G. Lorentz: Bernstein polynomials. Chelsea 1986, 140p.

G. Lorentz: Approximation of functions. Chelsea 1986. A classic.

1509 Manfred Mueller: Approximationstheorie. 
Akademische Verlagsgesellschaft 1978.

5772 S. Nikolskii: Approximation of functions of several variables and 
imbedding theorems. Springer 1975.

5720 Niels Erik Noerlund: Vorlesungen ueber Differenzenrechnung. Chelsea 1954.

G. Nuernberger: Approximation by spline functions.
Springer 1989, 243p. DM 74. A clear introduction to an interesting and 
developing field of numerical analysis.

George Phillips: Interpolation and approximation by polynomials.
Springer 2003, 310p.

A. Pinkus: On L{1}-approximation. Cambridge UP 1989, 240p. $45.

Manfred Reimer: Constructive theory of multivariate functions, with an 
application to tomography. Bibl. Inst. 1990, 290p. DM 38.

Manfred Reimer: Multivariate polynomial approximation. Birkhäuser 2003, 370p. Eur 98.

16012 Theodore Rivlin: The Chebyshev polynomials. Wiley 1974, 190p.

S. Saitoh: Integral transforms, reproducing kernels and their applications.
Chapman & Hall 1997, 280p. $100.

24414 Tomas Sauer: Approximationstheorie. Vorl. Univ. Gießen 2010, 190p.

16251 Robert Schaback: Native Hilbert spaces for radial basis functions I-II.
Internet 1996, 31p.

16252 Robert Schaback/Holger Wendland: Approximation by positive definite
kernels. Internet 2002, 13p.

24428 Robert Schaback: Approximationstheorie I.
Vorl. Univ. Göttingen 2010, 210p.

5769 Walter Schempp/Karl Zeller (ed.): Constructive theory of functions of 
several variables. SLN Math. 571 (1977).

H. Shapiro: Topics in approximation theory.
Springer LN Math. 187 (1971).

27120 David Sprecher: On the structure of continuous functions of several variables.
Trans. AMS 115 (1964), 340-355.

7630 Herbert Stahl: Best uniform rational approximation of x^\alpha
on [0,1]. Bull. AMS 28 (1993), 116-122.

Herbert Stahl/Vilmos Totik: General orthogonal polynomials.
Cambridge UP 1992, 250p. 0-521-41534-9. $ 60.

Karl-Georg Steffens: The history of approximation theory from Euler
to Bernstein. Birkhäuser 2006, 220p.

5564 H. Strauss: Review of the book "Approximation by spline functions" by 
G. Nuernberger. Jber. DMV 94 (1992), 47-49.

V. Tikhomirov: Convex analysis and approximation theory.
Springer 1990, 270p. DM 128. Seems to be a good, excellently written book.

A. Timan: Theory of approximation of functions of a real variable.
Dover 1994, 640p. $10.

24427 Rüdiger Verfürth: Approximationstheorie. Vorl. Univ. Bochum 2010, 140p.

8495 Jet Wimp: Review of the book "Polynomials and polynomial inequalities"
by Borwein/Erdelyi. Math. Intell. 18/3 (1996), 76-79.

24397 Gaetano Zanghirati: Metodi di approssimazione numerica. Internet 2013, 190p.