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On a polynomial inequality of Kolmogoroff's type

Author(s): B. D. Bojanov; A. K. Varma
Journal: Proc. Amer. Math. Soc. 124 (1996), 491-496.
MSC (1991): Primary 41A17
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Abstract: We prove an inequality of the form

$\begin{displaymath}\|f^{(j)}\|^2\leq A\|f^{(m)}\|^2+B\|f\|^2\end{displaymath}$

for polynomials of degree $n$ and any fixed $0<j<m\leq n$ . Here $\|\cdot\|$ is the $L_2$ -norm on $(-\infty,\infty)$ with a weight $e^{-t^2}$ . The coefficients $A$ and $B$ are given explicitly and depend on $j,m$ and $n$ only. The equality is attained for the Hermite orthogonal polynomials $H_n(t)$ .

References:

1: A. N. Kolmogorov, Inequalities between the upper bounds of consecutive derivatives of functions on the unbounded interval. Uchen. Zap. MGU Math. 30 (1939), 3--13. (Russian)
2: N. Korneichuk, Exact constants in approximation theory, Cambridge Univ. Press, Cambridge, 1991. MR 92m:41002
3: E. Schmidt, Über die nebstihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1943), 165--204. MR 6:212c
4: V. M. Tikhomirov, Some questions in approximation theory, Moscow State University, Moscow, 1976. (Russian) MR 58:6822
5: A. K. Varma, A new characterization of Hermite polynomials, Acta Math. Hungar. 49 (1987), 169--172. MR 88b:41016


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