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
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