Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

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
Full-text PDF Free Access

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. 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
  • 3 Erhard Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum, Math. Ann. 119 (1944), 165–204 (German). MR 0011754
  • 4 V. M. \cyr{T}ikhomirov, Nekotorye voprosy teorii priblizhenii, Izdat. Moskov. Univ., Moscow, 1976 (Russian). MR 0487161
  • 5 A. K. Varma, A new characterization of Hermite polynomials, Acta Math. Hungar. 49 (1987), no. 1-2, 169–172. MR 869673, 10.1007/BF01956321

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 41A17

Retrieve articles in all journals with MSC (1991): 41A17


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

DOI: https://doi.org/10.1090/S0002-9939-96-03024-9
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