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≤∞,
for all absolutely continuousf with\(||f'W||_{L_p (R)} \) finite? We show that such a theorem is true iff both
and
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 anL 1 but not anL ∞ Jackson theorem (or conversely).
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Research supported by NSF grant DMS-0400446 and US-Israel BSF grant 2004353.
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Lubinsky, D.S. Which weights on ℝ admit Jackson Theorems?admit Jackson Theorems?. Isr. J. Math. 155, 253–280 (2006). https://doi.org/10.1007/BF02773956
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DOI: https://doi.org/10.1007/BF02773956