Polynomial and rational approximation of continuous functions in $\mathbb{C}$

Question

I am wondering what the state of the art is for polynomial and rational approximations to continuous/holomorphic functions in $\mathbb{C}$. The particular domains of interest are the closed unit ball $\overline{\mathbb{D}}$ and the square $S:=[-1,1]+[-1,1]i$.

The first setting of greatest interest is as follows.

Given $f:\overline{\mathbb{D}}\to\mathbb{C}$ continuous along with precision $\varepsilon$, is there a function $c(f,\varepsilon)$, or even $c(f)$, such that one can compute $g\in\mathbb{C}[z]$ with each of the following holding?

1. $\|f-g\|_{L^\infty(\overline{\mathbb{D}})}<\varepsilon$,
2. $\|g\|_{L^\infty(S)}<c(f,\varepsilon)$ [that is, $g$ extends $f$ without blowing up], and
3. $\deg g$ is minimized (ideally $O(\log\frac{1}{\varepsilon})$ or $O(\mathrm{poly}\log\frac{1}{\varepsilon})$).

The second setting of greatest interest is to relax $g$ to just being continuous.

Given $f:\overline{\mathbb{D}}\to\mathbb{C}$ continuous along with precision $\varepsilon$, is there a function $c(f,\varepsilon)$, or even $c(f)$, such that one can compute $g\in\mathbb{C}[x,y]$ with each of the following holding?

1. $\|f(z)-g(\mathrm{re}z,\mathrm{im}z)\|_{L^\infty(\overline{\mathbb{D}})}<\varepsilon$,
2. $\|g(\mathrm{re}z,\mathrm{im}z)\|_{L^\infty(S)}<c(f,\varepsilon)$ [that is, $g$ extends $f$ without blowing up], and
3. $\deg g$ is minimized (ideally $O(\log\frac{1}{\varepsilon})$ or $O(\mathrm{poly}\log\frac{1}{\varepsilon})$).

For these two problems, the ideal would be if there is some plane version of (the constructive version of) Jackson's theorem, or the Remez algorithm, but each modified to facilitate extension.

A related problem is the following.

Given $f:\overline{\mathbb{D}}\to\mathbb{C}$ continuous along with precision $\varepsilon$, is there a function $\tilde{c}(f,\varepsilon)$, or even $\tilde{c}(f)$, such that one can compute $g,h\in\mathbb{C}[z]$ with each of the following holding?

1. $\left\|f-\frac{g}{h}\right\|_{L^\infty(\overline{\mathbb{D}})}<\varepsilon$,
2. $\inf\limits_{z\in S}\lvert h(z)\rvert>\tilde{c}(f,\varepsilon)$ [so that in scaling $g$ and $h$ by the same factor to make them well-bounded on $S$, we don't overcompensate by sending $h$ too close to 0], and
3. $\deg g$ and $\deg h$ are minimized (ideally $O(\log\frac{1}{\varepsilon})$ or $O(\mathrm{poly}\log\frac{1}{\varepsilon})$).

Is there anything effective known about any of these (or related) problems? I am having some trouble getting a full sense of the literature, as most results appear to focus on the case of $\mathbb{R}$. The wrinkle of $\overline{\mathbb{D}}\subsetneq S$ seems to also be something that causes this to diverge meaningfully from the literature.

For instance, for the latter, it appears that the Padé approximants accomplish (1) and (3) via rational functions (I've seen no commentary one way or the other towards (2)), but I haven't found an explicit statement of the vanishing of $L^\infty$ in terms of the numerator/denominator degree, and most of the discussion I see about them seems focused on $\mathbb{R}$ (though Scholarpedia and this survey do discuss $\mathbb{C}$, albeit without explicit bounds).

Also, unfortunately, this pretty comprehensive Encyclopedia of Math article (though mainly about existence results rather than effective ways to compute the approximations) is nearly 40 years out of date, so if there exists something analogous from the last 20-or-so years that could be very helpful as well.

Answer 1

The answers to the first series of questions are negative since a continuous function in general CANNOT be uniformly approximated by polynomials of $z$ uniformly on a set with non-empty interior: indeed a uniform limit of polynomials must be analytic.

Same with the questions of the 3-d series: since your rational functions cannot have poles on $D$, they can approximate on $D$ only an analytic function.

The answer to question 1 of the first series is positive by the usual Weierstrass theorem which says that every continuous function can be uniformly approximated by a polynomial (in two variables $x,y$.)