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.



Approximation theory on the disc
If the main interest lies in numerical methods designed to compute specifically with functions defined on a disk, the following recent manuscript may be useful:


<https://arxiv.org/pdf/1604.03061.pdf>


A few basic (theoretical) differences between Chebyshev polynomial approximations on an interval and on a disk are as follows:


the main feature of Chebyshev approximation by polynomials on an interval is the fact that the family of approximants, at each degree $n$, is spanned by a Chebyshev system, that is a set of continuous functions such that every nonzero linear combination has at most $n$ zeros on the interval. This implies existence and uniqueness of a best approximant $p\_{n}^{\*}$, and also a characterization of $p\_{n}^{\*}$ in terms of the error function which alternates in sign at $n+2$ extremal points of the interval. A study of general Chebyshev systems can be found in [2].


When approximating real functions on the unit disk by real polynomials in $x$ and $y$, the property of being a Chebyshev system is obviously lost, and thus also the ``Chebyshev theory''. Thus, different arguments are used:


* Existence of a best approximant follows from the continuity of the sup norm with respect to the coefficients of the polynomials.
* In general uniqueness of a best approximant is false. For instance, it is known that a best approximant to the monomial $x^{n}y^{m}$ in the disk by polynomials of total degree less than $n+m$ is unique if and only if $n=0$, $m=0$, or $n=m=1$.
* Characterization of a best approximant can be given. The notion of extremal signatures is sometimes used, see [4]. Such characterizations
can be seen as extensions of the alternation property that holds for the Chebyshev systems. They essentially follow from the Hahn-Banach separation theorem. For instance,
the classical Kolmogorov criterion ([4] p.65 or [1] p.10) gives the following characterization of a best uniform approximant from a convex set $M$ in $C(X)$ where $X$ is a compact set:


The function $u\_0\in M$ is a best uniform approximant to $f\in C(X)$ from $M$ iff
$$\forall u\in M,\quad\inf\_{x\in P[\epsilon\_0]}\text{Re}(\overline{\epsilon\_0(x)}(u(x)-u\_0(x)))\leq 0,$$
where $\epsilon\_0=f-u\_0$ and $P[\epsilon\_0]=\{x\in X,~|\epsilon\_0(x)|=\|\epsilon\_0\|\}$.
In the particular case when $M$ is an $n$-dimensional linear subspace of real-valued (resp. complex-valued) functions of $C(X)$, a characterization is as follows:


There are $m$ points $x\_i\in P[\epsilon\_0]$ and $m$ positive numbers $\theta\_i$, $m\leq n+1$ (resp. $m\leq 2n+1$) such that
$$\forall u\in M,\quad\sum\_{i=1}^m\theta\_i\overline{\epsilon\_0(x\_i)}u(x\_i)=0.$$
From a numerical point of view, practical implementations of Chebyshev approximation on a - *square* - are discussed in [5], considering approximation of a continuous function by expansions in terms of products $T\_{n}(x)T\_{m}(y)$, $n,m\geq 0$, where $T\_{n}$ denotes the univariate Chebyshev polynomials.


Finally, analogs of Chebyshev polynomials in the disk have also been studied in [3].


References


[1] D. Braess, Nonlinear approximation theory. Springer Series in Computational Mathematics, 7. Springer-Verlag, Berlin, 1986.


[2] S. Karlin, W.J. Studden, Chebyshev systems: with applications in analysis and statistics, Wiley, New York, 1966.


[3] I. Moale; F. Peherstorfer, Explicit min-max polynomials on the disc. J. Approx. Theory 163 (2011), no. 6, 707-723.


[4] T. Rivlin, Chebyshev polynomials,
From approximation theory to algebra and number theory, Pure and Applied Mathematics, Wiley, Inc., New York, 1990.


[5] A. Sommariva, M. Vianello, R. Zanovello, 
Adaptive bivariate Chebyshev approximation. 
Numer. Algorithms 38 (2005), no. 1-3, 79-94. 


