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The $L_p$ version of Newman's Inequality
for lacunary polynomials


Authors: Peter Borwein and Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 124 (1996), 101-109
MSC (1991): Primary 41A17; Secondary 30B10, 26D15
DOI: https://doi.org/10.1090/S0002-9939-96-03022-5
MathSciNet review: 1285974
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Abstract | References | Similar Articles | Additional Information

Abstract: The principal result of this paper is the establishment of the essentially sharp Markov-type inequality

\begin{equation*}\|xP^{\prime }(x)\|_{L_p[0,1]} \leq \left (1/p+12 \left({\sum ^n_{j=0}}(\lambda _j + 1/p)\right)\right) \|P\|_{L_p[0,1]} \end{equation*}

for every $P \in \text{span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ with distinct real exponents $\lambda _j$ greater than $-1/p$ and for every $p \in [1, \infty ]$. A remarkable corollary of the above is the Nikolskii-type inequality

\begin{equation*}\|y^{1/p}P(y)\|_{L_\infty [0,1]} \leq 13 \left({\sum ^n_{j=0}}(\lambda _j + 1/p)\right)^{1/p} \|P\|_{L_p[0,1]} \end{equation*}

for every $P \in \text{\rm span}\{x^{\lambda _0}, x^{\lambda _1}, \ldots , x^{\lambda _n}\}$ with distinct real exponents $\lambda _j$ greater than $-1/p$ and for every $p \in [1, \infty ]$. Some related results are also discussed.


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

  • 1. P. B. Borwein and T. Erdélyi, Polynomials and polynomials inequalities, Springer-Verlag, New York (to appear).
  • 2. ------, The full Müntz theorem in $C[0,1]$, $L_2[0,1]$, and $L_1[0,1]$, J. London Math. Soc. (to appear).
  • 3. ------, Müntz systems and orthogonal Müntz--Legendre polynomials, Trans. Amer. Math. Soc. 342 (1994), 523--542. MR 94f:42026.
  • 4. C. Frappier, Quelques problemes extremaux pour les polynomes at les functions entieres de type exponentiel, Université de Montréal, Québec, 1982.
  • 5. D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360--362. MR 55:3609

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-96-03022-5
Keywords: Müntz polynomials, lacunary polynomials, Dirichlet sums, Markov-type inequality, $L_p$ norm
Received by editor(s): June 28, 1994
Additional Notes: The research of the first author was supported, in part, by NSERC of Canada. The research of the second author was supported, in part, by NSF under Grant No. DMS-9024901 and conducted while an NSERC International Fellow at Simon Fraser University.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society

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