Which weights on \(\mathbb R\) admit Jackson theorems? (Q2472734)

Let \(W: \mathbb{R} \to (0, \infty)\) be a continuous function. The main result of this paper is that the following propositions are equivalent: (a) There exists a sequence \((\eta_n)_{n=1}^{\infty}\) of positive numbers with limit \(0\) such that for every \(1 \leq p \leq \infty\), \[ \inf_{\text{ deg}(P) \leq n}\| (f-P)W\| _{L_p(\mathbb{R})} \leq \eta_n\| f^{\prime}W\| _{L_p(\mathbb{R})} \] for all absolutely continuous \(f\) with \(\| f^{\prime}W\| _{L_p(\mathbb{R})}\) finite. (b) \[ \lim_{x \to \pm \infty} W(x) \int_0^x(W(t))^{-1}\, dt = 0 \quad \text{ and}\quad \lim_{x \to \pm \infty} (W(x))^{-1} \int_0^xW(t)\, dt = 0. \] It follows that (a) cannot be satisfied if \(W(x) = e^{-| x| }\). The author also shows that the first condition in (b) is necessary and sufficient for (a) to be satisfied for \(p=\infty\), and that the second condition in (b) is necessary and sufficient for (a) to be satisfied for \(p=1\), and constructs weights \(W(x)\) for which (a) is satisfied for \(p=1\) but not for \(p=\infty\), and weights for which (a) is satisfied for \(p=\infty\) but not for \(p=1\).