Some Sharp Estimates for the Haar System and Other Bases In L1(0,1)

Authors

  • Adam Osȩkowski

DOI:

https://doi.org/10.7146/math.scand.a-18006

Abstract

Let h=(hk)k0 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 , then \Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta, for any sequence \varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n of signs. A related bound for an arbitrary basis of L^1(0,1) is also established. The proof rests on the construction of the Bellman function corresponding to the problem.

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Published

2014-08-12

How to Cite

Osȩkowski, A. (2014). Some Sharp Estimates for the Haar System and Other Bases In L^1(0,1). MATHEMATICA SCANDINAVICA, 115(1), 123–142. https://doi.org/10.7146/math.scand.a-18006

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Articles