# A remark on approximation with polynomials and greedy bases

## TL;DR

This paper studies polynomial approximation errors with constant and modulus-constant coefficients in Banach spaces, characterizing their limits in relation to democracy properties of bases and extending previous results.

## Contribution

It provides new characterizations of approximation error limits using democracy functions and extends prior results on polynomial approximation in Banach spaces.

## Key findings

- Characterizes when approximation errors are equivalent to the norm.
- Provides conditions for the limits of approximation errors to equal the norm.
- Extends previous results on polynomial approximation with 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 Bern\'a and Blasco (2016) to study greedy bases in Banach spaces. We characterize when $\liminf_{m}{\mathcal{D}_{m}(x)}$ and $\liminf_{m}{\mathcal{D}_{m}^*(x)}$ are equivalent to $\| x\|$ 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.