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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: 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
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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

REFERENCES

[1]Alspach, D. and Odell, E.Lp Spaces. Handbook of the geometry of Banach spaces, Vol. I, 123–159 (North-Holland, 2001).CrossRefGoogle Scholar
[2]Dilworth, S. J., Kutzarova, D. and Temlyakov, V. N.Convergence of some greedy algorithms in Banach spaces. J. Fourier Anal. Appl. 8 (2002), 489–505.CrossRefGoogle Scholar
[3]Dilworth, S. J., Kutzarova, D., Shuman, K., Wojtaszczyk, P. and Temlyakov, V. N.Weak Convergence of greedy algorithms in Banach spaces. J. Fourier Anal. Appl. 14 (2008), 609–628.CrossRefGoogle Scholar
[4]Ganichev, M. and Kalton, N. J.Convergence of the weak dual greedy algorithm in Lp-spaces. J. Approx. Theory 124 (2003), 89–95.CrossRefGoogle Scholar
[5]Ganichev, M. and Kalton, N. J.Convergence of the dual greedy algorithm in Banach spaces. New York J. Math. 15 (2009), 73–95.Google Scholar
[6]Huber, P. J.Projection pursuit. Ann. Statist. 13 (1985), 435–475.Google Scholar
[7]Jones, L.On a conjecture of Huber concerning the convergence of projection pursuit regression. Ann. Statist. 15 (1987), 880–882.CrossRefGoogle Scholar
[8]Livshits, E. D.Convergence of greedy algorithms in Banach spaces. Math. Notes 73 (2003), 342–358.CrossRefGoogle Scholar
[9]Lindenstrauss, J. and Tzafriri, L. Classical Banach spaces II, function spaces. Ergebnisse der Mathematik. 97 (Springer-Verlag, 1979).Google Scholar
[10]Temlyakov, V. N.Weak greedy algorithms. Adv. Comput. Math. 12 (2000), 213–227.CrossRefGoogle Scholar
[11]Temlyakov, V. N.Greedy algorithms in Banach spaces. Adv. Comput. Math. 14 (2001), 277–292.CrossRefGoogle Scholar
[12]Temlyakov, V. N.Nonlinear methods of approximation. Found. Comput. Math. 3 (2003), 33–107.CrossRefGoogle Scholar
[13]Temlyakov, V. N.Relaxation in greedy approximation. Constr. Approx. 28 (2008), 1–25.CrossRefGoogle Scholar