| Résume | In arxiv:1901.01339, we gave a counterexample to a Mergelyan type statement for cartesian products stated by Gamelin and Garnett in an article published in TAMS fifty years ago, in 1969. In order to correct this false statement we introduced a natural algebra $A_D(K)$, smaller than the classical algebra $A(K)$, where $K$ is a compact subset of $\Bbb C^d$. The introduction of $A_D(K)$ explains partially why approximation theory in several variables is not so developed and opens a new road of research. We believe that several new Mergelyan type theorems will be obtained. So far we have one such theorem for products and one for graphs. Applications to universality can be found in arxiv:1909.03521 and arxiv:1910.01759. Also, a parallel theory can be developed when replacing uniform convergence by uniform on $K$ convergence of all orders derivatives (in preparation). |
