We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
We establish a bijection between Dirichlet series and formal power series through Bohr’s transform. This is one of the main tools all along the text and relies on the fact that by the fundamental theorem of arithmetic every natural number has a unique decomposition as a product of prime numbers. In this way, to each such number a multi-index can be assigned (and vice-versa). With this we show that the space of bounded holomorphic functions on B_{c0} and \mathcal{H}_\infty are isomorphic as Banach spaces. This means that to every holomorphic function corresponds a Dirichlet series in such a way that the monomial and the Dirichlet coefficients are identified. We consider m-homogenous Dirichlet series: those having non-zero coefficients only if n has exactly m prime divisors (counted with multiplicity) and show that the space of such Dirichlet series is isometrically isomorphic to the space of m-homogeneous polynomials on c0.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
