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 $\fG_m(f,D)\>Cm^{1/2}\lvert\{f}\rvert_{H}$ 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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