Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ
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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.
KeywordsOrthogonal Polynomial Polynomial Approximation Exponential Weight Constructive Approximation BERNSTEIN Theorem
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- D. S. Lubinsky,Ideas of weighted polynomial approximation on (−∞, ∞), inApproximation and Interpolation (C. K. Chui and L. L. Shumaker, eds.), World Scientific, Singapore, 1995, pp. 371–396.Google Scholar
- D. S. Lubinsky,Which weights on ℝ admit Jackson Theorems?, Israel Journal of Mathematics, to appear.Google Scholar
- D. S. Lubinsky and E. B. Saff,Strong Asymptotics for Extremal Polynomials Associated with Exponential Weights, Lecture Notes in Mathematics1305, Springer, Berlin, 1988.Google Scholar