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 2 - Advanced Toolbox
Published online by Cambridge University Press: 19 July 2019
We continue the study initiated in Chapter 7 of polynomials with small norms. This time the norm of the polynomial is not taken as the supremum on the n-dimensional polydisc, we take it on B_X, the unit ball of some Banach space. The goal is to show that, given a polynomial, signs can be found in such a way that the norm of the new polynomial, whose coefficients are the original ones multiplied by the signs, has small norm. We do this with three different approaches. The first two approaches use Rademacher random variables as the main probabilistic tools. The first one is based on finding out how many balls of a fixed radius are needed to cover B_X while the second one uses entropy integrals and a good estimate for the entropy numbers of the inclusions between l_p spaces. The third approach is different, and relies on Gaussian random variables, Slepian’s lemma and the fact that Rademacher averages are dominated by Gaussian averages. This approach also allows to get estimates for vector-valued polynomials.
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.
