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A Lipschitz continuous function $f : [-1,1] \to \mathbb{C}$ has a unique representation as a series in terms of the Chebyshev polynomials $T_k$, $$ f(x) = \sum_{k = 0}^\infty a_k \, T_k(x) \qquad \forall x \in [-1,1] . $$ See e.g. Theorem 3.1 in Trefethen's "Approximation Theory and Approximation Practice".

Is there an analogue of this result for arbitrary compact sets $S \subset \mathbb{C}$? That is, given such an $S$, is there a sequence of polynomials $T_k^{(S)}$ such that every Lipschitz-continuous function $f : S \to \mathbb{C}$ can be written as $$ f(x) = \sum_{k = 0}^\infty a_k \, T_k^{(S)}(x) \qquad \forall x \in S ? $$ I'm actually only interested in the case of $f$ being analytic in a neighbourhood of $S$, so if that makes a difference then never mind the Lipschitz-continuous case.


What I have so far:

Given $f$ analytic in a neighbourhood of $S$, potential theory tells us that polynomial approximation converges exponentialy, i.e. $$ \min_{p_k \in \mathcal{P}_k} \|f - p_k\|_{\infty,S} \leq C \, \exp(-\gamma \, k) $$ for some $C, \gamma > 0$. Given this, it is tempting to take any polynomial basis $p_k$ and look at the limit $$ \DeclareMathOperator*{\argmin}{arg\,min} a = \lim_{n \to \infty} \argmin_{a^{(n)} \in \mathbb{C}^n} \left\|f - \sum_{k = 0}^n a_k^{(n)} p_k \right\|_{\infty,S}, $$ but this doesn't even exist in general, as outlined in my previous question here.

This example shows that we must be careful about the choice of the basis $p_k$. The Fekete polynomials as defined in Saff's "Logarithmic Potential Theory with Applications to Approximation Theory" look like a candidate because they generalise Chebyshev polynomials in the sense that they are the monic polynomials with asymptotically the smallest extremum. I don't see how this relates to the above problem though.


Why I am interested in this:

Given a matrix $A$ and a function $f$, I would like to estimate the size of the elements of $f(A)$. If I had the desired result, I could write $$ f(A)_{ij} = \sum_{k = 0}^\infty a_k \, T_k^{(S)}(A_{ij}) $$ and would be done if I can get both a bound on $a_k$ (probably from a polynomial approximation estimate) and a bound on $p_k(A_{ij})$ (by assuming certain decay properties of $A$). If $f$ is defined on a single interval, I can do exactly that using the theory on Chebyshev functions, but I am now interested in the case when the domain of $f$ consists of two or more intervals.

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  • $\begingroup$ If the series converges locally uniformly, the limit would be analytic in the interior of $S$. $\endgroup$ – Pietro Majer Nov 29 '16 at 21:58
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    $\begingroup$ The Chebyshev polynomials don't look relevant since they span the space of traditional polynomials. If you are asking about polynomial approximation of holomorphic functions, then Runge's theorem provides the answers: en.wikipedia.org/wiki/Runge's_theorem $\endgroup$ – Christian Remling Nov 30 '16 at 0:58
  • $\begingroup$ Runge's theorem doesn't give me more than what I already have: for every function $f$ holomorphic on a neighbourhood of the compact set $S$, there exists a sequence of polynomials approximating $f$. What I am interested in, however, is not an arbitrary sequence, but one which is a series in terms of known basis functions. I'll update the question to give some background on why I am interested in this problem. $\endgroup$ – gTcV Nov 30 '16 at 10:20
  • $\begingroup$ I don't get what you know and what you want to show... Can you state your question clearly in 5 lines ? $\endgroup$ – reuns Nov 30 '16 at 13:30
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    $\begingroup$ @user1952009 I think the question is clearly and concisely described in the first block. Can you be more specific as to what is unclear? $\endgroup$ – gTcV Nov 30 '16 at 13:36
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The generalization of Chebyshev series on $[-1,1]$ to an arbitrary connected and simply connected compact set $K$ of the complex plane were defined and studied by Faber in 1903. The series of Chebyshev polynomials corresponding to $[-1,1]$ are then replaced with series of Faber polynomials of the set $K$. These series play in $K$ the role of the Taylor series in the unit disk $\mathbb{D}$, and enjoy exactly the same convergence properties, which can be seen via the conformal map from $\overline{\mathbb{C}}\setminus\overline{\mathbb{D}}$ to $\overline{\mathbb{C}}\setminus K$. A standart reference for Faber polynomials is [1].

The case when $K$ has several components (like, for instance, the union of two intervals) was first considered by Walsh in 1958. Based on conformal mappings from multiply connected regions to lemniscates, that he also introduced, see [2], he defined in [3] the so-called Faber-Walsh polynomials which again have properties in $K$ similar to Taylor series. The papers [4] and [5] are recent references on that subject. In particular the case of two intervals is studied in detail in [5] along with numerical results.

[1] P. K. Suetin, Series of Faber Polynomials, Gordon and Breach, Amsterdam, 1998.

[2] J. L. Walsh, On the conformal mapping of multiply connected regions, Trans. Amer. Math. Soc., 82 (1956), pp. 128-146.

[3] J. L. Walsh, A generalization of Faber's polynomials, Math. Ann., 136 (1958), pp. 23-33.

[4] O. Sète, J. Liesen, On conformal maps from multiply connected domains onto lemniscatic domains. Electron. Trans. Numer. Anal. 45 (2016), 1-15.

[5] O. Sète, J. Liesen, Properties and examples of Faber-Walsh polynomials. Comput. Meth. Funct. Theory 17 (2017), 151-177.

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