Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C10

Retrieve articles in all journals with MSC (2000): 42C10


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

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08720-X
PII: S 0002-9939(06)08720-X
Received by editor(s): May 10, 2005
Published electronically: June 27, 2006
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2006 American Mathematical Society