Holomorphic Functions
Identifieur interne : 000B03 ( Istex/Checkpoint ); précédent : 000B02; suivant : 000B04Holomorphic Functions
Auteurs : János Horváth [États-Unis]Source :
- Bolyai Society Mathematical Studies [ 1217-4696 ] ; 2006.
Abstract
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.
Url:
DOI: 10.1007/978-3-540-30721-1_10
Affiliations:
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<front><div type="abstract" xml:lang="en">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.</div>
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