On the completeness of polynomials in a weighted space of entire functions
Let \(w\) be a weight function defined on \(L_\rho = \{z : |\arg z |= {\pi \over 2 \rho}, \rho \geq 1\}\) and let \({\mathcal C}_w (L_\rho)\) denote the space of all continuous functions on \(L_\rho\) satisfying the condition \(f(t)/w(t) \to 0\) as \(|t |\to \infty\), \(t \in L_\rho\). The topology in this space is given by the norm \(|f |= \sup_{t \in L_\rho} |f(t) |/w(t)\). Let \({\mathcal P}_w (L_\rho)\) be the closure in \({\mathcal C}_w (L_\rho)\) of the polynomials and let \({\mathcal C}^*_w (L_\rho)\) denote the set of entire functions of order \(\rho\) and of minimal type belonging to \({\mathcal C}_w (L_\rho)\). The main result of this paper is the following theorem: If \(w(t) = \exp (a |t |^\alpha)\), \(0 < \alpha < \rho/(2 \rho - 1)\), \(t \in L_\rho\), \(a > O\), then \({\mathcal P}_w (L_\rho) = {\mathcal C}^*_w (L_\rho)\).
scientific article; zbMATH DE number 795463