Sharpness of the QOGA Lebesgue-type inequality in Banach spaces

Determine whether the Lebesgue-type inequality \(\|f_m\|_X \le C(1+m)\sigma_m(f)_X\) for the Quasi-Orthogonal Greedy Algorithm in an \(M\)-coherent Banach-space dictionary is sharp.

Background

The paper studies the Quasi-Orthogonal Greedy Algorithm (QOGA), a Banach-space analogue of the Orthogonal Greedy Algorithm or Orthogonal Matching Pursuit. For an $M$-coherent dictionary, Corollary \ref{QOC0.2} gives the bound $\|f_m\|_X \le C(1+m)\sigma_m(f)_X$ for the residual after $m$ iterations, where $\sigma_m(f)_X$ is the best $m$-term approximation error in the Banach-space norm.

The unresolved issue is whether the factor $1+m$ has the optimal growth in $m$. The paper contrasts this with the Hilbert-space estimate in Corollary \ref{QOC2}, whose factor $1+m^{1/2}$ is stated to be sharp, while the corresponding Banach-space estimate remains unverified.