Operators and divergent series

• Autores: Richard Stone
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 217, Nº 2, 2004, págs. 331-374
• Idioma: inglés
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• Resumen

  • We give a natural extension of the classical de.nition of C¿esaro convergence of a divergent sequence/function. This involves understanding the spectrum of eigenvalues and eigenvectors of a certain C¿esaro operator on a suitable space of functions or sequences. The essential idea is applicable in identical fashion to other summation methods such as Borel's. As an example we how to obtain the analytic continuation of the Riemann zeta function æ(z) for Re z ¡Ü 1 directly from generalised C¿esaro summation of its divergent de.ning series. We discuss a variety of analytic and symmetry properties of these generalised methods and some possible further applications.