Holomorphic perturbation of Fourier coefficients (Q2701629)

Recently \textit{H. Render} [Proc. Am. Math. Soc. 127, No. 5, 1409-1411 (1999; Zbl 0915.46044)] proved:NEWLINENEWLINENEWLINEIf \(f(z)= \sum^\infty_{n=-\infty} a_nz^n\in H^\infty(\mathbb{D})\), the space of bounded analytic functions on the open disc \(\mathbb{D}\), and \(F\) is holomorphic in a neighbourhood \(U\) of \(0\) with \(F(0)= 0\) an \(a_n\in U\) \((n\in\mathbb{Z})\), then \(\sum^\infty_{n=-\infty} F(a_n) z^n\in H^\infty(\mathbb{D})\).NEWLINENEWLINENEWLINEThe author showed that this result follows from the abstract theory of Segal algebras, and that this approach allows to generalize Render's result to a large class of spaces of functions.