1. W.A.J. Luxemburg, "Müntz-Szász type approximation results and the Paley-Wiener Theorem". in Approximation Theory II, G.G. Lorentz, C.K. Chui and L.L. Schumaker eds.; pp. 437-448; Academic Press, 1976.
2. R. Siegmund-Schultze, Der Beweis des Weierstraßschen Approximationssatzes 1885 vor dem Hintergrund der Entwicklung der Fourieranalysis. (German) [The proof of Weierstrass' approximation theorem of 1885 against the background of the development of Fourier analysis] Historia Math. 15 (1988), no. 4, 299--310.
3. K. Weierstrass, Ausgewählte Kapitel aus der Funktionenlehre. (German) [Selected chapters from function theory] Vorlesung, gehalten in Berlin 1886, mit der akademischen Antrittsrede, Berlin 1857, und drei weiteren Originalarbeiten von K. Weierstraß aus den Jahren 1870 bis 1880/86. [Lecture, held in Berlin, 1886, with the inaugural address, Berlin, 1857, and three other original works by K. Weierstrass from the years 1870 to 1880/86]. With a preface by Kurt-R. Biermann. Edited and with a foreword, comments and an appendix by R. Siegmund-Schultze. With English, French and Russian summaries. Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], 9. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1988. 272 pp. ISBN: 3-322-00478-3
4. The method of Bourbaki (Dieudonne) and Sz.-Nagy for approximating $\sqrt{x}$ are really the same.
5.
W. Sierpinski, Dowód elementarny twierdzenia Weierstrass'a i wzoru
interpolacyjnego Borel'a, Prace Mat-fiz, t. XXII, (1911), 59-68. This
paper, which seems to have been overlooked, was pointed out to me by Milan
Jovanovic. E. Borel, in his book Lecons sur les fonctions de variables
reelles et les developpements en series de polynomes, Gauthier-Villars,
Paris, 1905, on p. 79-82, suggests an approximation procedure using
values of f in the following way. Approximate f by
\sum_{p=0}^q f(p/q) P_{p,q}
where P_{p,q} are polynomials (independent of f, and of undetermined degree)
obtained by approximating to the order 1/q^2 the function \phi_{p,q} which
is the hat function, zero off [(p-1)/q, (p+1)/q], piecewise continuous
with knots at i/q and 1 at p/q. No construction is given for P_{p,q},
rather the Weierstrass theorem is evoked to claim existence thereof. Borel
also says that it would be interesting to effectively calculate the
P_{p,q}, at least for small values of p and q. Sierpinski uses this
outline, as given by Borel, and explicitly constructs P_{p,q} without
invoking Weierstrass as Borel does. In this way, he also is proving the
Weierstrass Theorem. See the reference to p. 80 in Borel by Sierpinski. In
formula (7), Sierpinski gets a polynomial approximation to |x| on an
interval (previously he had a rational approximation). In (9), he gets an
approximation to the hat function, and the rest follows. He does not
mention Lebesgue who also had an approximation to |x|, but without error
estimates. In any case, all this was superseded one year later by the
appearance of Bernstein's paper where he introduced the Bernstein
polynomials.