Abstract
For the weights exp (−|x|λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP n can decrease on [−1,1] ifP n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL p spaces.
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Communicated by George G. Lorentz.
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Nevai, P., Totik, V. Weighted polynomial inequalities. Constr. Approx 2, 113–127 (1986). https://doi.org/10.1007/BF01893420
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DOI: https://doi.org/10.1007/BF01893420