Distance to finite degree polynomials for BV functions

Question

A result of Jackson establishes for lipschitz functions $f\in\text{W}^{1,\infty}(0,1)$ the bound $$\inf_{p\in\mathbf{R}_n[x]} \|f-p\|_\infty\lesssim \frac{1}{n}\|f'\|_\infty,$$ where $\mathbf{R}_n[x]$ denotes the set of polynomial functions of degree $\leq n$, the infinite norms being taken over $(0,1)$.

Generalizations exist for Sobolev regularity replacing the $\|\cdot\|_\infty$ by $\|\cdot\|_p$, possibly in higher dimension, see for instance the Bramble-Hilbert Lemma.

My question is (in the simplest case of dimension 1) : is there a known estimate for $f\in \text{BV}(0,1)$ of the following form

$$\inf_{p\in\mathbf{R}_n[x]} \|f-p\|_1\lesssim \delta_n \|f'\|_{\text{TV}},$$ where $\|\cdot\|_{\text{TV}}$ stands for the total variation (of course $\delta_n$ is expected to converge to $0$) ?

Answer 1

Thanks to Giorgio's comment I found the good reference. In fact De Vore and Lorentz give a refined estimate (Theorem 6.1, Chapter 7) in comparison with the Bramble-Hilbert Lemma I've just cited : $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_p \lesssim \omega_p(f,\frac{1}{n}),$$ where $\omega_p(f,\cdot)$ is the modulus of continuity of $f$ for the $\text{L}^p$ norm. Since any element of $\text{W}^{1,p}(0,1)$ satisfies $\omega_p(f,\delta) \lesssim_f \delta $, so we recover the Bramble-Hilbert cases. In fact, the previous estimate on the modulus of continuity characterizes the corresponding Sobolev space inside $\text{L}^p(0,1)$ (the constant behind $\lesssim_f$ being $\|f'\|_p)$, except for $p=1$, for which this characterizes $\text{BV}(0,1)$ (with constant $\|f'\|_{\text{TV}}$ and not only $\text{W}^{1,1}(0,1)$. In particular, we have therefore, for $f\in\text{W}^{1,1}(0,1)$ $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_1 \lesssim \frac{1}{n}\|f'\|_{\text{TV}}.$$