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\mytitle{Lebesgue-type inequalities in greedy approximation with respect to bases}

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\author{
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	Jos\'e L. Ansorena \\ %
    University of La Rioja \\ 
    \url{joseluis.ansorena@unirioja.es}
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}


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Let $(\GG_m)_{m=1}^\infty$ denote the thresholding greedy algorithm (TGA for short) of a basis of a Banach space $\XX$. To measure the efficiency of the TGA is customary to use the Lebesgue parameters $(\leb_m)_{m=1}^\infty$, defined for each $m\in\NN$ as the optimal constant $C=C(m)$ such that 
\[
\norm{f-\GG_m(f) }\le C \norm{f -g}
\]
for all $f\in\XX$ 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 $(\leb_m)_{m=1}^{\infty}$. Most of of such parameters and estimates have sprung from the celebrated characterization of greedy bases  by Konyagin and Telmyakov  \cite{KoTe1999}. 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 $(\unc_m)_{m=1}^\infty$  and a sequence of democracy-like parameters that  fits their purposes.

In this talk, we introduce a new sequence of democracy-like parameters, which we call $(\sq_m)_{m=1}^\infty$, which combined linearly with the unconditionality parameters determines the growth of the Lebesgue parameters. That is,
\[
\leb_m\approx \max\{ \unc_m,\sq_m\}, \quad m\in\NN.
\] 
 
This result provides an answer to a problem raised by Temlyakov during the \emph{Concentration week on greedy
 algorithms in Banach spaces and compressed sensing} held on 18--22 July, 2011, at Texas A\&M University.


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\paragraph{Joint work with:} Fernando Albiac, Pablo Bern\'a.


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\bibitem{KoTe1999}
S.~L.~Konyagin and V.~N.~Temlyakov.
\newblock A remark on greedy approximation in Banach spaces.
\newblock {\em East J. Approx.}, 5(3):365--379, 1999.

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