Abstract
In this article we prove that for any orthonormal system (\varphi_j)_{j=1}^n \subset L_2 that is bounded in L_{\infty} , and any 1 < k < n , there exists a subset I of cardinality greater than n-k such that on \mathrm{span}\{\varphi_i\}_{i \in I} , the L_1 norm and the L_2 norm are equivalent up to a factor \mu (\log \mu)^{5/2} , where \mu = \sqrt{n/k} \sqrt{\log k} . The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
