Lebesgue constants for Chebyshev thresholding greedy algorithms

Abstract

We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L ch n for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L ch n with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained. MSC: 41A25; 41A46; 42A10

Lebesgue inequalities for Chebyshev thresholding greedy algorithms

“…These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.…”

“…Our first result is a general upper bound, which improves and extends [18,Theorem 2.4]. Theorem 1.1.…”

“…This completes the proof of Lemma 3.4, and hence of Theorem 1.3.Remark 3.5. When B is a Schauder basis, a similar proof gives the following lower bound, which is also obtained in[18, Theorem 2.2]…”

“…The proof, however, seems to contain a gap, and a missing factor k c m should also appear in the last summand. Nevertheless, it is still fair to ask whether the inequality (3.7) asserted in [18] may be true with a different proof. Remark 3.2.…”

“…The second result is a slight generalization of [9,Theorem 4.1], and gives a correct version of [18,Theorem 3.5]. Theorem 1.2.…”