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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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