Which weights on ℝ admit Jackson Theorems?admit Jackson Theorems?

Abstract

Let W: ℝ→(0,∞) be continuous. DoesW admit a classical Jackson Theorem? That is, does there exist a sequence\(\{ \eta _n \} _{n = 1}^\infty \) of positive numbers with limit 0 such that for 1≤p≤∞, $$\mathop {\inf }\limits_{\deg (P) \le n} ||(f - P)W||_{L_p (R)} \le \eta n||f'W||_{L_p (R)} $$ for all absolutely continuousf with\(||f'W||_{L_p (R)} \) finite? We show that such a theorem is true iff both $$\mathop {\lim }\limits_{\chi \to \infty } W(\chi )\int_0^\chi {W^{ - 1} } = 0$$ and $$\mathop {\lim }\limits_{\chi \to \infty } W^{ - 1} (\chi )\int_\chi ^\infty W = 0,$$ with analogous limits asx→−∞. In particular,W(x)=exp(−|x|) does not admit a Jackson theorem of this type. We also construct weights that admit anL1 but not anL∞ Jackson theorem (or conversely).