# Pure greedy algorithm

I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that $$||f-G_m(f,D)||>Cm^{-1/2}|f|_{H}$$ for some constant $C$ i.e. convergence rate is bigger than $-1/2$? I know that there are such dictionaries that CR is bigger than $-1/2$( even $-0.27$) but I can not construct such a Riesz basis. My steps, that might be helpful.There is a theorem that if $$\sup_{g\in D }\sum_{g'\in D,g'\not=g}|\langle g,g'\rangle|<1/3$$ than convergence rate is less than $-1/2$.That is why I took Riesz basis for which that supremum equal to 1: $$D=(f_j), f_j=e_j+\frac{1}{2}e_{j+1}$$ Now I want to find $f$. But I can not :(

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What is $A_1(D)$? – Davide Giraudo Nov 4 '12 at 12:03
I added it, sorry that is was not clear – StudentMath Nov 4 '12 at 12:22
Thanks. And $G_m(f,D)$? – Davide Giraudo Nov 4 '12 at 12:36
$G_m(f,D)$ is approximant that we obtain at $m$th step of pure greedy algorithm – StudentMath Nov 4 '12 at 12:39