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

Nikolskii-type inequalities for shift invariant function spaces

Authors: Peter Borwein and Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 134 (2006), 3243-3246
MSC (2000): Primary 41A17
DOI: https://doi.org/10.1090/S0002-9939-06-08533-9
Published electronically: June 6, 2006
MathSciNet review: 2231907
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Let $V_{n}$ be a vectorspace of complex-valued functions defined on ${\mathbb {R}}$ of dimension $n+1$ over ${\mathbb {C}}$. We say that $V_{n}$ is shift invariant (on ${\mathbb {R}}$) if $f \in V_{n}$ implies that $f_{a} \in V_{n}$ for every $a \in {\mathbb {R}}$, where $f_{a}(x) := f(x-a)$ on ${\mathbb {R}}$. In this note we prove the following.

Theorem. Let $V_{n} \subset C[a,b]$ be a shift invariant vectorspace of complex-valued functions defined on ${\mathbb {R}}$ of dimension $n+1$ over ${\mathbb {C}}$. Let $p \in (0,2]$. Then \begin{equation*}\|f\|_{L_{\infty }[a + \delta ,b-\delta ]} \leq 2^{2/p^{2}} \left (\frac {n+1}{\delta } \right )^{1/p} \|f\|_{L_{p}[a,b]} \end{equation*} for every $f \in V_{n}$ and $\delta \in \left (0,\frac {1}{2}(b-a) \right ) .$

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Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: terdelyi@math.tamu.edu

Keywords: Nikolskii-type inequalities, shift invariant function spaces, exponential sums
Received by editor(s): May 17, 2005
Published electronically: June 6, 2006
Communicated by: David Preiss