Some Sharp Estimates for the Haar System and Other Bases In L1(0,1)
DOI:
https://doi.org/10.7146/math.scand.a-18006Abstract
Let h=(hk)k≥0 denote the Haar system of functions on [0,1]. It is well known that h forms an unconditional basis of Lp(0,1) if and only if 1<p<∞, and the purpose of this paper is to study a substitute for this property in the case p=1. Precisely, for any λ>0 we identify the best constant β=βh(λ)∈[0,1] such that the following holds. If n is an arbitrary nonnegative integer and a0, a1, a2, …, an are real numbers such that ‖∑nk=0akhk‖1≤1, then |{x∈[0,1]:|n∑k=0εkakhk(x)|≥λ}|≤β, for any sequence ε0,ε1,ε2,…,εn of signs. A related bound for an arbitrary basis of L1(0,1) is also established. The proof rests on the construction of the Bellman function corresponding to the problem.Downloads
Published
2014-08-12
How to Cite
Osȩkowski, A. (2014). Some Sharp Estimates for the Haar System and Other Bases In L1(0,1). MATHEMATICA SCANDINAVICA, 115(1), 123–142. https://doi.org/10.7146/math.scand.a-18006
Issue
Section
Articles