Approximations on $\mathcal{RH}_\infty$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:27:43Z http://mathoverflow.net/feeds/question/50323 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50323/approximations-on-mathcalrh-infty Approximations on $\mathcal{RH}_\infty$ kosti 2010-12-25T04:59:33Z 2010-12-25T04:59:33Z <p>Hi there, I have checked it but I hope I am not missing a previously asked question. It goes as such:</p> <p>I am reading some material about convex optimization and there is a particular part that I couldn't get a handle on it. </p> <p>[...]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. $P(x) = \begin{pmatrix}1 &amp;\frac{1}{x+a} &amp;\frac{1}{(x+a)^2} &amp;\cdots&amp;\frac{1}{(x+a)^{n-1}}\end{pmatrix}^T$, $x\in\mathbb{C}_0$,$a>0$ and a matrix $L\in\mathbb{R}^{n\times m}$ such that </p> <p>$\displaystyle \lim_{n\to\infty}\|L^TP - f\|_\infty = 0$</p> <p>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 $\epsilon$. </p> <p>How can we show this or can anyone direct me to a less complicated sources on the subject?</p> <p>My apologies in advance if the question is too trivial to be asked here. </p>