Abstract
In 1978, Freud, Giroux and Rahman established a weightedL 1 Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL 1. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL p Jackson theorem for exp(−|x|), valid in allL p , 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing rates of approximation in terms of smoothness of the approximated function.
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Research supported by NSF grant DMS 0400446.
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Lubinsky, D.S. Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ. Isr. J. Math. 153, 193–219 (2006). https://doi.org/10.1007/BF02771783
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DOI: https://doi.org/10.1007/BF02771783