Convergence of greedy approximation for the trigonometric system |
| |
Authors: | S V Konyagin V N Temlyakov |
| |
Institution: | (1) Department of Mechanics and Mathematics, Moscow State University, Moscow, 19992, Russia;(2) Department of Mathematics, Univesity of South Carolina, Columbia, SC, 29208, U.S.A. |
| |
Abstract: | 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 ΛZd 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=1∞ to 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 ║f - Gm(f )║p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|