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Fast decreasing polynomials. (English) Zbl 0682.41014
Summary: Matching two-sided estimates are given for the minimal degree of polynomials P satisfying \(P(0)=1\) and \(| P(x)| \leq \exp (-\phi (| x|))\), \(x\in [-1,1]\), where \(\phi\) is an arbitrary, in [0,1], increasing function. Besides these fast decreasing polynomials we also consider bell-shaped polynomials and polynomials approximating well the signum function.

MSC:
41A10 Approximation by polynomials
26C05 Real polynomials: analytic properties, etc.
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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[1] A. L. Levin, D. S. Lubinsky (1987):Canonical products and the weights exp(x|\(\alpha\)),\(\alpha\)>l,with applications. J. Approx. Theory,49:149–169. · Zbl 0619.41006 · doi:10.1016/0021-9045(87)90085-2
[2] A. L. Levin, D. S. Lubinsky (1987):Weights on the real line that admit good relative polynomial approximation, with applications. J. Approx. Theory,49:170–195. · Zbl 0612.41010 · doi:10.1016/0021-9045(87)90086-4
[3] D. S. Lubinsky (1985):Estimates of Freud-Christoffel functions for some weights with the whole real line as support. J. Approx. Theory,44:343–379. · Zbl 0584.42015 · doi:10.1016/0021-9045(85)90086-3
[4] P. Nevai, V. Totik (1986):Weighted polynomial inequalities. Constr. Approx.,2:113–127. · Zbl 0604.41014 · doi:10.1007/BF01893420
[5] E. B. Saff, V. Totik (1989):Polynomial approximation of piecewise analytic functions. J. London Math. Soc.,39:487–498. · Zbl 0683.41011 · doi:10.1112/jlms/s2-39.3.487
[6] Bl. Sendov (1979): Hausdorff Approximation. Sofia: Bulgarian Academy of Sciences. · Zbl 0463.34007
[7] P. Turān (1968):On an inequality of Chebyshev. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.,11:15–16. · Zbl 0172.05901
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