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On a polynomial inequality
of Kolmogoroff's type

Authors: B. D. Bojanov and A. K. Varma
Journal: Proc. Amer. Math. Soc. 124 (1996), 491-496
MSC (1991): Primary 41A17
MathSciNet review: 1291763
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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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Additional Information

B. D. Bojanov
Affiliation: Department of Mathematics, University of Sofia, Blvd. James Boucher 5, 1126 Sofia, Bulgaria
Email: BOR@BGEARN.bitnet

A. K. Varma
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611

Received by editor(s): January 3, 1994
Received by editor(s) in revised form: August 25, 1994
Additional Notes: The first author was supported in part by the Bulgarian Ministry of Science under Grant No. MM-414
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society