This paper deals with radial weights on balanced domains \(G\subset \mathbb{C}^ n\), for which it is possible to apply methods involving the Taylor series of holomorphic functions about zero. The contractive properties of the Cesàro means of the Taylor series of functions in the disk algebra are used to derive remarkable consequences for spaces of holomorphic functions on arbitrary balanced open sets in \(\mathbb{C}^ n\). This leads to simple proofs that the spaces \(HV_ 0(G)\) and \({\mathcal V}_ 0 H(G)\) have the bounded approximation property whenever they contain the polynomials, and that then the polynomials are dense.
The second part of the paper is devoted to a related problem on \(\varepsilon\)-tensor products with an arbitrary Banach space with applications to certain spaces of vector-valued holomorphic functions. In the last part some remarkable vector-valued generalizations of the (bi- )dualities \(((HV_ 0(G)_ b')_ b'= HV(G)\) and \((({\mathcal V}_ 0 H(G))_ b')_ i'= {\mathcal V} H(G)\) are established.