Quadratic convexity (Q1283051)

The classical definition of lineally convex sets in \(\mathbb{C}^n\) and of the Fantappiè transform uses the pairing \(\langle z,\zeta \rangle= \zeta (z)\) where \(\zeta\) is a polynomial of degree 1 with the normalization \(\zeta (0)= 1\). In this paper, the author replaces polynomials of degree 1 by polynomials of degree 2 and defines the notion of quadratic convexity and of quadratic Fantappiè transform \({\mathcal Q}\mu\) of an analytic functional \(\mu\). He then gives examples of compact quadratically convex sets for which any \(f\in {\mathcal O}(K)\) can be written \({\mathcal Q}\mu\). The proof gives an explicit formula for such a functional \(\mu\) satisfying \({\mathcal Q}\mu=f\) based on a Cauchy-Fantappiè-Leray type integral representation of \(f\).