Cyclicity of non vanishing functions in the polydisc and in the ball
Pascal Thomas (Institute de Mathématiques de Toulouse, Université Paul Sabatier)

Summary. Consider a Banach space \(X\) of holomorphic functions in the polydisc or ball, defined by weighted integral conditions, or more generally by weighted sums of powers of the coefficients of the Taylor expansion of the functions in \(X\). Assume that \(X\) contains all the functions bounded and holomorphic on the relevant domain. We give a condition on the space \(X\) (roughly speaking that its norm be small enough) so that any nonvanishing bounded function \(f\) in \(X\) will be cyclic, i.e. any other \(g\) in \(X\) can be approximated by \(P_n f\), where \((P_n)\) is a sequence of polynomials. To do this, we use a special version of the Corona Theorem in several variables, valid when all but one of the data functions are smooth. This generalizes results obtained by El Fallah, Kellay and Seip in the case of disc, themselves motivated by classical results of Nikolai Nikolski.

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