Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the closed unit disc? I can only find papers which discuss analytic complex valued functions, which is not what I need.

UPDATE: Some basic features of Chebyshev theory are as follows. Suppose we have a continuous function $f$ on $[-1,1]$ (which may be expensive to compute). Let $P_nf$ be the unique polynomial of degree $n$ that agrees with $f$ at the Chebyshev points $\cos(k\pi/n)$ for $0\leq k\leq n$. There is an efficient procedure for computing $P_nf$, which is tolerant of rounding errors. It does not involve any integration: we just need to evaluate $f$ at $n+1$ points. There are various results about the convergence of $P_nf$ to $f$, assuming various hypotheses about the smoothness or real-analyticity of $f$. In typical examples, convergence is rapid. There is an extensive theory surrounding all this, in which the Chebyshev polynomials $T_n(x)=\cos(n\,\cos^{-1}(x))$ play a central role.

It is known that everything works much worse if we use equally spaced points instead of Chebyshev points. This is closely related to the Bernstein approximation system mentioned by Liviu Nicolaescu. That has some attractive theoretical properties, but usually converges quite slowly. Similarly, even if $f$ is analytic, the polynomials $P_nf$ typically converge to $f$ more rapidly than the Taylor approximations.