Universal entire functions with gap power series
The following result on ``lacunary universal functions'' is proved: Let \({\mathcal M}\) be the family of all compact sets \(K\) in \(\mathbb{C}\) which have connected complement and let \(A(K)\) denote the Banach space of all functions which are continuous on \(K\) and holomorphic in its interior \(K^\circ\). Theorem 1. Let \(Q\) be any subsequence of \(\mathbb{N}_0\) with density \(\Delta(Q)= 1\) and let \(\{z_n\}\) be any unbounded sequence of complex numbers. Then there exists an entire function \(\phi\) with lacunary power series \(\phi(z)= \sum^\infty_{v= 0}\phi_v z^v\), \(\phi_v= 0\) for \(v\not\in Q\) which is ``universal'' in the double sense that (A) the sequence \(\{\phi(z+ z_n)\}\) is dense in \(A(K)\), (B) the sequence \(\{\phi(zz_n)\}\) is dense in \(A(K)\) if \(0\not\in K\). The proof uses essentially the Riesz representation theorem and the Hahn-Banach theorem. Further results concern an investigation of the sharpness of the density property of \(Q\).
scientific article; zbMATH DE number 1251211
scientific article