Let (G_m) denote the thresholding greedy algorithm (TGA for short) of a basis of a Banach space X. To measure the efficiency of the TGA is customary to use the Lebesgue parameters (L_m), defined for each positive integer m as the optimal constant C such that 

||f-G_m(f)|| ≤ C ||f -g||

for all vectors, f, in X and all linear combinations, g, of m vectors of the basis.

Calculating the exact value of the Lebesgue constants can be in general a difficult task, so in order to study the efficiency of non-greedy bases, we must settle for obtaining easy-to-handle parameters that control the asymptotic growth of (L_m). Most of such parameters and estimates have sprung from the celebrated characterization of greedy bases by Konyagin and Telmyakov. In fact, several authors have obtained estimates for the Lebesgue constants, either of general bases or of bases with some special features, in terms of the unconditionality constants $(k_m)$ and a sequence of democracy-like parameters that fits their purposes.

In this talk, we introduce a new sequence of democracy-like parameters which combined linearly with the unconditionality parameters determines the growth of the Lebesgue parameters.

This result provides an answer to a problem raised by Temlyakov during the Concentration week on greedy

 algorithms in Banach spaces and compressed sensing held on 18--22 July 2011, at Texas A&M University.