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Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

Article contents

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: dilworth@math.sc.edu
E. ODELL
Affiliation:
Department of Mathematics, The University of Texas, 1 University Station C1200, Austin, TX 78712, U.S.A. e-mail: odell@math.utexas.edu
TH. SCHLUMPRECHT
Affiliation:
Department of Mathematics, Texas A & M University, College Station, TX 78743, U.S.A. e-mail: schlump@math.tamu.edu
ANDRÁS ZSÁK
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF. Peterhouse College, Cambridge, CB2 1RD. e-mail: A.Zsak@dpmms.cam.ac.uk
Corresponding
E-mail address:
dilworth@math.sc.edu
odell@math.utexas.edu
schlump@math.tamu.edu
A.Zsak@dpmms.cam.ac.uk
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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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