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Greedy approximation with respect to certain subsystems of the Walsh orthonormal system

Authors: Martin G. Grigorian and Robert E. Zink
Journal: Proc. Amer. Math. Soc. 134 (2006), 3495-3505
MSC (2000): Primary 42C10
Published electronically: June 27, 2006
MathSciNet review: 2240661
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Abstract: In an article that appeared in 1967, J.J. Price has shown that there is a vast family of subsystems of the Walsh orthonormal system each of which is complete on sets of large measure. In the present work it is shown that the greedy algorithm, when applied to functions in $ L^{1}[0,1]$, is surprisingly effective for these nearly-complete families. Indeed, if $ \Phi $ is such a subsystem of the Walsh system, then to each positive $ \varepsilon $, however small, there corresponds a Lebesgue measurable set $ E$ such that for every $ f$, Lebesgue integrable on $ [0,1]$, the greedy approximants to $ f$, associated with $ \Phi $, converge, in the $ L^{1}$ norm, to an integrable function $ g$ that coincides with $ f$ on $ E$.

References [Enhancements On Off] (What's this?)

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  • 2. M.G. Grigorian and Robert E. Zink, Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for $ L^{p}[0,1],1\leq p<+\infty $, Proc. Amer. Math. Soc. 131(4) (2002), 1137-1149. MR 1948105 (2003k:42052)
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Additional Information

Martin G. Grigorian
Affiliation: Department of Physics, Erevan State University, Alex Manoogian Str., 375049 Yerevan, Armenia

Robert E. Zink
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1968

Received by editor(s): May 10, 2005
Published electronically: June 27, 2006
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society

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