Approximation properties of some types of sequences of entire functions (Q1594386)

Let \(H(\mathbb{C})\) be the space of all entire functions with the topology defined by uniform convergence on compact sets. Let \(H(\mathbb{C})^*\) be the space of all linear functionals over the space \(H(\mathbb{C})\), then we can identify \(H(\mathbb{C})^*\) with the class of all functions that are analytic in a neighborhood of infinity and such that \(\gamma (\infty)=0\). The topology in \(H(\mathbb{C})^*\) is defined by uniform convergence on one of the circles \(|z|=r\), \(0<r< \infty\). A system \(\{\varphi_n\}\) in \(H(\mathbb{C})\) (or \(H(\mathbb{C})^*)\) is called complete if the closure of its linear hull is \(H(\mathbb{C})\) (or \(H(\mathbb{C})^*)\). Let \(f\in H(\mathbb{C})\) and \(\{h_n\}\) be a complete system in \(H(\mathbb{C})^*\). It is known that if \(f\) is not a polynomial, then the sequence of the function \(f_n(z)= (f(z)h_n (z))^+\) is complete in \(H(\mathbb{C})\), here \(u^+(z)\) means the nonnegative part of the Laurent series \(u(z)\). If \(f\) is a polynomial, the system \(\{f_n\}\) is incomplete in \(H(\mathbb{C})\). In this paper, the author refines the above result for special systems \(\{h_n\}\). The author proves that (some other consequences are discussed in more details from this) for \(h_n(z)=z^{-n}(1+ \sum^\infty_{k=1} b_{k,n} z^{-k})\), \(n>0\), with the sequence of coefficients \(\{b_{k,n}\}\) satisfying the estimate \(|b_{k,n} |\leq cR^k\) for all \(k,n>0\), where \(c,R>0\) are constants, if \(f\in H(\mathbb{C})\) is not a polynomial, then for any nonnegative function \(\Phi\) defined on the set of natural numbers, there always exists a strictly increasing sequence of positive integers \(\{n_s\}\) such that (i) \(n_{s+1}> \Phi(n_s)\) for all \(s>0\); (ii) the sequence \(\{f_{n_s} (z)\}\) is complete in \(H(\mathbb{C})\); moreover, if \(E_p=\text{span}\{f_{n_s}\}\), \(q(q+1)/2\leq s\leq(q+1) (q+2)/2\), \(q\geq 0\) \((\dim(E_q) =q+1)\), then each of the spaces \(E_q\) contains a function \(g_q\) such that the system \(\{g_q(z) \}^\infty_{q=0}\) forms a quasi-polynomial basis in \(H(\mathbb{C})\).