Greedy approximation with respect to certain subsystems of the Walsh orthonormal system
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- by Martin G. Grigorian and Robert E. Zink PDF
- Proc. Amer. Math. Soc. 134 (2006), 3495-3505 Request permission
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
- M.G. Grigorian, On the convergence of the greedy algorithm in the $L^{1}$ norm (to appear).
- 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 (2003), no. 4, 1137–1149. MR 1948105, DOI 10.1090/S0002-9939-02-06618-2
- T. W. Körner, Decreasing rearranged Fourier series, J. Fourier Anal. Appl. 5 (1999), no. 1, 1–19. MR 1682270, DOI 10.1007/BF01274186
- J. J. Price, A density theorem for Walsh functions, Proc. Amer. Math. Soc. 18 (1967), 209–211. MR 209760, DOI 10.1090/S0002-9939-1967-0209760-7
- V. N. Temlyakov, Greedy algorithm and $m$-term trigonometric approximation, Constr. Approx. 14 (1998), no. 4, 569–587. MR 1646563, DOI 10.1007/s003659900090
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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 3495-3505
- MSC (2000): Primary 42C10
- DOI: https://doi.org/10.1090/S0002-9939-06-08720-X
- MathSciNet review: 2240661