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Analysis Mathematica

, Volume 31, Issue 2, pp 85–115 | Cite as

Convergence of greedy approximation for the trigonometric system

  • S. V. Konyagin
  • V. N. Temlyakov
Article

Summary

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f  we take as an approximant a trigonometric polynomial of the form  Gm(f ) := ∑kЄΛ f^(k) e (i k,x), where ΛˆZ d  is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that Gm(f ) gives the best m-term approximant in the L2-norm and, therefore, for each f ЄL2,  ║f-Gm(f )║2→0  as  m →∞.  It is known from previous results that in the case of p ≠2 the condition f ЄLp does not guarantee the convergence  ║f-Gm(f )║p→0  as  m →∞..  We study the following question. What conditions (in addition to f ЄLp) provide the convergence  ║f-Gm(f )║p→0  as  m →∞?  In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {An}n=1to guarantee the Lp-convergence of {Gm(f )} for all f ЄLp  , satisfying an (f ) ≤An , where {an (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄLp and  ║GM(m)(f ) - Gm(f )║p  →0  as  m →∞ imply  ║fGm(f )║p  →0  as  m →∞. We have found these conditions in the case when p is an even number or p = ∞.

Keywords

Periodic Function Fourier Coefficient Trigonometric Polynomial Nonlinear Method Greedy Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag/Akadémiai Kiadó 2005

Authors and Affiliations

  • S. V. Konyagin
    • 1
  • V. N. Temlyakov
    • 2
  1. 1.Department of Mechanics and Mathematics, Moscow State University
  2. 2.Department of Mathematics, Univesity of South Carolina

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