Meilleure approximation polynomiale et croissance des fonctions entières sur certaines variétés algébriques affines. (Best polynomial approximation and growth of entire functions on certain affine algebraic varieties)
Let K be a polynomially convex compact set in \({\mathbb{C}}^ n\) and \(V_ K\) its logarithmic extremal potential in \({\mathbb{C}}^ n\). Suppose that K is regular (e.g. \(V_ K\) is continuous). Then we construct for every holomorphic function f on K a sequence \((P_{\ell})_{\ell \geq 1}\) of polynomials in n complex variables with deg (P\({}_{\ell})\leq \ell\) such that: \(\forall \vartheta >1\), \(\exists c_{\vartheta}>0\), \(\exists N_{\vartheta}>1\) such that \[ \max_{z\in K} | f(z)- P_{\ell}(z)| \leq [c_{\vartheta}(1+r)^{N_{\vartheta}}/(r-\quad 1)^{2n-1}]\cdot \| f\|_{{\bar \Omega}_{r\vartheta}}/r^{\ell} \] for every \(r>1\), \(\ell \geq 1\), f holomorphic on \({\bar \Omega}{}_{r\vartheta}\) where \(\Omega_{r\vartheta}=\{z\in {\mathbb{C}}^ n: \exp V_ K(z)<r\vartheta \}.\) The constants \(c_{\vartheta},N_{\vartheta}\) don't depend on r,\(\ell\) and f. Moreover, \(P_{\ell}\) depends linearly on f and is given by an integral operator obtained from the theory of integral formula with weight kernels of Khenkin-Ramirez type developed recently by Berndtsson and Anderson. The above estimates are then used to extend to several complex variables a classical result of S. N. Bernstein relating the analytic continuation of a continuous function on K by an entire function of prescribed order and type with its best polynomial approximation on K.
integral formulas with weight kernels of Henkin-Ramirez type
analytic continuation of continuous functions by entire functions
scientific article; zbMATH DE number 3959941
scientific article