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On the convergence of greedy algorithms for initial segments of the Haar basis

Published online by Cambridge University Press:  15 January 2010

S. J. DILWORTH ,
E. ODELL ,
TH. SCHLUMPRECHT  and
ANDRÁS ZSÁK
S. J. DILWORTH
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. e-mail: [email protected]
E. ODELL
Affiliation:
Department of Mathematics, The University of Texas, 1 University Station C1200, Austin, TX 78712, U.S.A. e-mail: [email protected]
TH. SCHLUMPRECHT
Affiliation:
Department of Mathematics, Texas A & M University, College Station, TX 78743, U.S.A. e-mail: [email protected]
ANDRÁS ZSÁK
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF. Peterhouse College, Cambridge, CB2 1RD. e-mail: [email protected]
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Abstract

We consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 3 , May 2010 , pp. 519 - 529
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

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