Jackson and Bernstein theorems for the weight exp(-vertical bar x vertical bar) on R

摘要

In 1978, Freud, Giroux and Rahman established a weighted L-1 Jackson theorem for the weight exp(-|x|) on the real line, using methods that work only in L-1. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|(alpha)), alpha >= 1, where weighted polynomials are dense, and the case alpha < 1, where weighted polynomials are not dense. We obtain the first L-p Jackson theorem for exp(-|x|), valid in all L-p, 0 < p <= infinity, 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.