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 | 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 , is surprisingly effective for these nearly-complete families. Indeed, if is such a subsystem of the Walsh system, then to each positive , however small, there corresponds a Lebesgue measurable set such that for every , Lebesgue integrable on , the greedy approximants to , associated with , converge, in the norm, to an integrable function that coincides with on .

**1.**M.G. Grigorian,*On the convergence of the greedy algorithm in the norm*(to appear).**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≤𝑝<+∞*, Proc. Amer. Math. Soc.**131**(2003), no. 4, 1137–1149. MR**1948105**, 10.1090/S0002-9939-02-06618-2**3.**T. W. Körner,*Decreasing rearranged Fourier series*, J. Fourier Anal. Appl.**5**(1999), no. 1, 1–19. MR**1682270**, 10.1007/BF01274186**4.**J. J. Price,*A density theorem for Walsh functions*, Proc. Amer. Math. Soc.**18**(1967), 209–211. MR**0209760**, 10.1090/S0002-9939-1967-0209760-7**5.**V. N. Temlyakov,*Greedy algorithm and 𝑚-term trigonometric approximation*, Constr. Approx.**14**(1998), no. 4, 569–587. MR**1646563**, 10.1007/s003659900090

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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

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08720-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