Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-bcmtx Total loading time: 7.45 Render date: 2021-04-18T09:54:53.481Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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
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

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
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below.

References

[1]Alspach, D. and Odell, E.Lp Spaces. Handbook of the geometry of Banach spaces, Vol. I, 123159 (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), 489505.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), 609628.CrossRefGoogle Scholar
[4]Ganichev, M. and Kalton, N. J.Convergence of the weak dual greedy algorithm in Lp-spaces. J. Approx. Theory 124 (2003), 8995.CrossRefGoogle Scholar
[5]Ganichev, M. and Kalton, N. J.Convergence of the dual greedy algorithm in Banach spaces. New York J. Math. 15 (2009), 7395.Google Scholar
[6]Huber, P. J.Projection pursuit. Ann. Statist. 13 (1985), 435475.CrossRefGoogle Scholar
[7]Jones, L.On a conjecture of Huber concerning the convergence of projection pursuit regression. Ann. Statist. 15 (1987), 880882.CrossRefGoogle Scholar
[8]Livshits, E. D.Convergence of greedy algorithms in Banach spaces. Math. Notes 73 (2003), 342358.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), 213227.CrossRefGoogle Scholar
[11]Temlyakov, V. N.Greedy algorithms in Banach spaces. Adv. Comput. Math. 14 (2001), 277292.CrossRefGoogle Scholar
[12]Temlyakov, V. N.Nonlinear methods of approximation. Found. Comput. Math. 3 (2003), 33107.CrossRefGoogle Scholar
[13]Temlyakov, V. N.Relaxation in greedy approximation. Constr. Approx. 28 (2008), 125.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 18 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 18th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the convergence of greedy algorithms for initial segments of the Haar basis
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the convergence of greedy algorithms for initial segments of the Haar basis
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the convergence of greedy algorithms for initial segments of the Haar basis
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *