In this thesis we solve the coboundary equation $delta c=d$ with bounds for cochains with values in a coherent subsheaf of some $mathcal{O}^p_{Omega}$, where $Omega$ is a Stein manifold. In particular the existence of a finite set of global generators is not assumed. Our result applies therefore to the ideal sheaf $mathcal{J}_Vsubset mathcal{O}_{C^N}$ of germs of holomorphic functions vanishing on a closed analytic submanifold $VsubsetC^N$. Although we are mainly interested in the estimates for the solutions of $delta c=d$, the techniques used also lead to a proof for the classical Theorem B of Cartan for coherent subsheafs of some $mathcal{O}^p_{Omega}$, avoiding the Mittag-Leffler argument. We derive an extension theorem for holomorphic functions on V to entire functions, with control on growth behaviour. ewline As a corollary we construct a linear tame extension operator $H(V)to H(C^N)$ under the hypothesis that H(V) is linear tamely isomorphic to the infinite type power series space $Lambda_infty(k^{frac{1}{n}})$, n= dim$_{C}V$; this condition is also necessary. Here the supnorms on H(V) are taken over intersections of V with polycylinders of polyradii e^m, $min N$. Aytuna asked how much, and what kind of, information about the complex analytic structure of V is carried by the Frechet space H(V). We prove that H(V) is linear tamely isomorphic to a power series space of infinite type if and only if V is algebraic.
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