I study pure greedy algorithms in different bases. I am interested in the following 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 every constant $C$? In other words, is there a Riesz basis that yields a convergence rate bigger than $-1/2$?
I know that there are such dictionaries whose convergence rate is larger than $-1/2$ (even $-0.27$) but I cannot construct such a Riesz basis.
Note: A Pure Greedy Algorithm is defined inductively by \begin{align*} G_0(f) & = 0 \\ G_m(f) & = G_{m-1}(f)+G(f-G_{m-1}(f)) \end{align*} where $G(f)=\langle f,g\rangle g$ and $g=g(f)$ is an element from $D$ that maximizes $\left|\langle f,g\rangle\right|$.
Here are some ideas that might be helpful. It is known that if $$ \sup_{g\in D} \sum_{g'\in D,~ g'\ne g} \left|\langle g,g'\rangle\right| < 1/3, $$ then the convergence rate is less than $-1/2$. That is why I took Riesz basis for which that supremum is equal to 1: $$ D=(f_j), ~\text{where}~ f_j=e_j+\frac{1}{2}e_{j+1}. $$ Now I want to find $f$. But I can not :(
Some useful information can be found in this paper by DeVore and Temlyakov and this paper by Temlyakov.