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.

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