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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 :(

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 $\|f-G_m(f,D)\|>Cm^{-1/2}\lvert\{f}\rvert_{H}$ 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 :(

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 $f\in A_1(D)$ such that $$||f-G_m(f,D)||>Cm^{-1/2}|f|_{A_1(D)}$$ 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.

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 :(

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 $f\in A_1(D)$ such that $$||f-G_m(f,D)||>Cm^{-1/2}|f|_{A_1(D)}$$ 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. Here, for some $M>0$ we define $$B_1(D,M)=\{f\in H: f=\sum_{k\in L} c_kg_k, g_k\in D,|L|<\infty, \sum_{k\in L}|c_k|<M\}$$ $A_1(D,M)$ is closure of $B_1(D,M)$ in $H$, and finally $A_1(D)$ is union of these $A_1(D,M)$s over all $M>0$

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 $f\in A_1(D)$ such that $$||f-G_m(f,D)||>Cm^{-1/2}|f|_{A_1(D)}$$ 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.