Analysis Mathematica

, Volume 31, Issue 2, pp 85–115

# 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
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