Holomorphic Functions

Abstract

It is quite well known that, as Jean-Pierre Kahane tells us elsewhere in this volume, Hungarian mathematics started the twentieth century with a bang when in October 1900 a twenty year old student, who had just returned from Berlin after a year there, proved that the Fourier series of a continuous function is uniformly Cesàro-summable to the function. It is, however, less well known that Lipót Fejér has already previously published an article containing some simple theorems concerning power series (Mat. Fiz. Lapok 9 (1900), 405–410; [40], No. 1, p. 29). A typical result is the following: if g is a positive integer and f an entire function of genus ≤ g − 1, then the radius of convergence of $\sum\nolimits_{n = 0}^\infty {c_n f\left( n \right)x^{n^g } }$ is not smaller than the the radius of convergence of $\sum\nolimits_{n = 0}^\infty {c_n x^{n^g } }$. An application is the fact that a power series and the series obtained from it by termwise differentiation have the same radius of convergence.