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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a polynomial inequality of Kolmogoroff’s type
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by B. D. Bojanov and A. K. Varma PDF
Proc. Amer. Math. Soc. 124 (1996), 491-496 Request permission

Abstract:

We prove an inequality of the form \[ \|f^{(j)}\|^2\leq A\|f^{(m)}\|^2+B\|f\|^2\] 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
  • 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)
  • N. Korneĭchuk, Exact constants in approximation theory, Encyclopedia of Mathematics and its Applications, vol. 38, Cambridge University Press, Cambridge, 1991. Translated from the Russian by K. Ivanov. MR 1124406, DOI 10.1017/CBO9781107325791
  • Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
  • V. M. Tikhomirov, Nekotorye voprosy teorii priblizheniĭ, Izdat. Moskov. Univ., Moscow, 1976 (Russian). MR 0487161
  • A. K. Varma, A new characterization of Hermite polynomials, Acta Math. Hungar. 49 (1987), no. 1-2, 169–172. MR 869673, DOI 10.1007/BF01956321
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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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 491-496
  • MSC (1991): Primary 41A17
  • DOI: https://doi.org/10.1090/S0002-9939-96-03024-9
  • MathSciNet review: 1291763