Polynomial approximation conjecture for active exterior acoustic cloaking

Let ${\bf c}^*=(\beta,0)$ be as in the proof of Theorem~. For $L>0$, define

$$D_{\beta,L}=\left\{z\in\mathbb{C}:|z-\beta|^L|z|<\frac{\beta^{L+1}L^L}{(L+1)^{L+1}}\right\}.$$

Let $S_1$ be any disk in the connected component of $D_{\beta,L}$ containing the origin, and let $S_2$ be any disk in the connected component of $D_{\beta,L}$ containing ${\bf c}^*$. For every $\epsilon>0$, there exist positive integers $s$ and $n$ such that

$$\left|\frac{s}{n}-L\right|<\epsilon$$

and the polynomial

$$P_{n,s}(z)=\left(1-\frac{z}{\beta}\right)^s\sum_{j=0}^{n-1}\left(\frac{z}{\beta}\right)^j\binom{s+j-1}{j}$$

satisfies

$$|P_{n,s}-1|<\epsilon\quad\text{on }\partial S_1, \qquad |P_{n,s}|<\epsilon\quad\text{on }\partial S_2.$$

Moreover, the approximation property referred to in the source is not satisfied when either $S_1$ or $S_2$ is not contained in $D_{\beta,L}$. Polynomial approximation conjecture. The stated approximation property holds precisely for disks contained in the specified connected components of $D_{\beta,L}$, and fails when either disk is not contained in $D_{\beta,L}$. This conjecture would extend the authors' earlier explicit polynomial cloaking results by allowing greater freedom in the location and size of the cloaked region; the source reports numerical evidence but does not provide a proof.