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While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this recent question for the background.

It should be useful to know the best approximation for some specific functions, one of which is the modulus function, $f(x)=|x|$. This function could be of great importance for estimating the distance between two complex variables. Thus it could be utilized as subroutines in solving other problems. My question is what is the best approximation to the modulus function in a given domain such as the unit disk $x\in \mathbb{C},|x|\leq 1$? Some references to this problem would be very welcome.

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    $\begingroup$ $|x|$ is always real, which a polynomial cannot be. So it is not clear to me how this should work. $\endgroup$
    – gmvh
    Feb 2 at 9:01
  • $\begingroup$ The intention was to say that the variable has a bounded modulus value, say $x\in\mathbb{C}$ and $|x|\leq 1$. I have modified the question accordingly. $\endgroup$ Feb 2 at 12:18
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    $\begingroup$ The modulus maps the unit disk into the unit interval. A polynomial won't do that, so I still don't get how the approximation is meant to work. Polynomials are holomorphic, the modulus isn't. $\endgroup$
    – gmvh
    Feb 2 at 13:49
  • $\begingroup$ @gmvh I think OP is asking for polynomials $f$ with low $\sup_{z \in D} |f(z) - |z||$, where the inner $|\cdot| : \mathbb{C} \to \mathbb{C}$ (that is, its codomain is treated as $\mathbb{C}$ rather than $\mathbb{R}$ via the canonical injection of the latter into the former). $\endgroup$
    – user76284
    Feb 2 at 22:07
  • $\begingroup$ @gmvh Note that OP is asking for approximations, so it doesn't matter that $|\cdot|$ itself isn't holomorphic. $\endgroup$
    – user76284
    Feb 2 at 22:07

1 Answer 1

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Let $\mathbb{D} \subset \mathbb{C}$ be the unit disc. Let $\phi : (\mathbb{D} \to \mathbb{C}) \to \mathbb{R}$ be the following functional:

$$ \phi = f \mapsto \sup_{z \in \mathbb{D}} |f(z) - |z|| $$

The question asks for polynomials $f$ with low $\phi$.

Partial answer: By Morera's theorem, the uniform limit of holomorphic functions on an open set of $\mathbb{C}$ is holomorphic.

$|\cdot|$ is not holomorphic.

Therefore, there is no sequence of holomorphic (e.g. polynomial) functions that converge uniformly to $|\cdot|$ on an open set of $\mathbb{C}$ (e.g. $\mathbb{D}$).

Therefore, $\phi$ cannot be arbitrarily small. Thus $\inf_f \phi(f) > 0$.

A trivial upper bound is $\inf_f \phi(f) \leq \phi(z \mapsto 0) = 1$.

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    $\begingroup$ The use of Morera's theorem is both overkill and only gives a qualitative negative result. Instead, note that the normalized integral of any holomorphic function $h:{\bf D} \to {\bf C}$ around the circle $z=re^{i\theta}$, for fixed $0<r<1$, is zero, while the normalized integral of $|z|$ round this contour is $r$. So in fact, the distance of $|z|$ from the disc algebra is $1$, and this tells us that all holomorphic approximations are as bad as just approximating the function by zero. $\endgroup$
    – Yemon Choi
    Feb 3 at 13:06

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