A remark on approximation with polynomials and greedy bases

Abstract

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 Berná and Blasco (2016) to study greedy bases in Banach spaces. We characterize when ${\mathrm{liminf}}_m{\mathcal{D}}_m(x)$ and ${\mathrm{liminf}}_m{\mathcal{D}}_m^{\ast }(x)$ are equivalent to $\Vert x\Vert$ in terms of the democracy and superdemocracy functions, and provide sufficient conditions ensuring that ${\lim }_m{\mathcal{D}}_m^{\ast }(x)={\lim }_m{\mathcal{D}}_m(x)=\Vert x\Vert$, extending previous very particular results.