A remark on approximation with polynomials and greedy bases

We investigate properties of the $m$-th error of approximation by polynomials with constant coefficients ${\mathcal{D}}_m(x)$ and with modulus-constant coefficients ${\mathcal{D}}_m^{\ast }(x)$ introduced by Bern'a and Blasco (2016) to study greedy bases in Banach spaces. We characterize when ${\mathrm{lim\hspace{0.17em}inf}}_m{\mathcal{D}}_m(x)$ and ${\mathrm{lim\hspace{0.17em}inf}}_m{\mathcal{D}}_m^{\ast }(x)$ are equivalent to $\mathrm{\parallel }x\mathrm{\parallel }$ in terms of the democracy and superdemocracy functions, and provide sufficient conditions ensuring that $\lim_{m}{$\mathcal{D}${m}^*(x)} = \lim{m}{$\mathcal{D}$_{m}(x)} = | x|$, extending previous very particular results.