Lebesgue inequalities for Chebyshev thresholding greedy algorithms

Abstract

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. 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. 1991 Mathematics Subject Classification. 41A65, 41A46, 46B15.

Characterization of Weight-Semi-greedy Bases

“…Here, we present a weaker condition than Schauder to give a characterization of semi-greediness<, that is, the version of Theorem 1.11 for the weight w = (1, 1, ...). For that purpose, we consider the following definition that we can find in [4]. Of course, this condition is satisfied for Schauder bases, but, in fact, it is satisfied for a more general bases.…”

“…Some examples of bases that are not Schauder satisfying the above proposition can be found in Section 3 of [4]. One of them is the trigonometric system in C([0, 1]).…”

“…Question: Recently, in [4], the authors proved that for Schauder bases and w = (1, 1, ...), the constants of super-democracy and disjoint-super-democracy are of the same order up to the basis constant, that is, C sd ≤ C s ≤ 2(K b + 1)C sd + κK b , where κ = sup n e n e * n . Is it possible to show the same result for general weights?…”

Strong partially greedy bases and Lebesgue-type inequalities

“…For further information on the subject we refer the reader to the monograph by V. N. Temlyakov [15] and to the articles [9,14,16,17] and the reference therin. For Lebesgue-type inequalities see, for instance, the more recent papers [4,6,10,12,13]). A greedy operator of order m is any mapping G m : X → X such that…”

“…Among the definitions given above, the parameters g m , g c m , gm , q m , γ m , L m , L m and L re m are well-known in the literature and were studied, for instance, in [2,4,6,9,10,17].…”