Approximations on $\mathcal{RH}_\infty$

2010-12-25T04:59:33Z

Hi there, I have checked it but I hope I am not missing a previously asked question. It goes as such:

I am reading some material about convex optimization and there is a particular part that I couldn't get a handle on it.

[...]Suppose we are given a real rational function $f \in \mathcal{RH}_\infty:\mathbb{C}_0\to\mathbb{C}^m$ which is bounded on the imaginary axis. Then, one can find a uniform approximation on the imaginary axis with a finite length basis e.g. $P(x) = \begin{pmatrix}1 &\frac{1}{x+a} &\frac{1}{(x+a)^2} &\cdots&\frac{1}{(x+a)^{n-1}}\end{pmatrix}^T$, $x\in\mathbb{C}_0$,$a>0$ and a matrix $L\in\mathbb{R}^{n\times m}$ such that

$\displaystyle \lim_{n\to\infty}\|L^TP - f\|_\infty = 0 $

The reference cited is Allan Pinkus' "n-width Approximation Theory" and it is beyond my understanding. But apparently there is a, seemingly innocent, density result that says for any $\epsilon$ there exists such sufficiently large $n$ and $a$ for the infinity norm of the error less than $\epsilon$.

How can we show this or can anyone direct me to a less complicated sources on the subject?

My apologies in advance if the question is too trivial to be asked here.

Approximations on $\mathcal{RH}_\infty$ - MathOverflow

Approximations on $\mathcal{RH}_\infty$