Conditions for the convergence of sequences in certain spaces of analytic functions (Q788862)

Let F(z) be an entire function and all its Taylor coefficients \^F(n) be different from zero. It is well-known that the space A'\({}_ R\) which is dual to the space \(A_ R\) of the functions analytic in the disc \(| z|<R\) can be identified with the space of the entire functions G(z) such that \[ \lim \sup_{n\to \infty}| \hat G(n)/\hat F(n)|<R. \] The sequence \(G_ k(z)\) in such A'\({}_ R\) is called convergent to zero if \[ \lim_{k\to \infty}| \hat G_ k(n)/\hat F(n)| R_ 1^{-n}=0 \] uniformly with respect to n for some \(R_ 1\in(0,R).\) The author obtains the necessary and sufficient condition on the function F(z) providing that the mentioned convergence is equivalent to the convergence defined by the maximum modulus of functions \(G_ k(z)\) and F(z): \[ \lim_{k\to \infty}(\max_{| z| =r}| G_ k(z)|)/(\max_{| z| =r}| F(R_ 1z)|)=0 \] uniformly with respect to \(r\in [0,\infty)\) for some \(R_ 1\in(0,R).\)