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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
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
Published electronically: June 6, 2006
MathSciNet review: 2231907
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Abstract: 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

$\displaystyle \Vert f\Vert _{L_{\infty }[a + \delta ,b-\delta ]} \leq 2^{2/p^{2}} \left (\frac{n+1}{\delta } \right )^{1/p} \Vert f\Vert _{L_{p}[a,b]} $

for every $ f \in V_{n}$ and $ \delta \in \left (0,\frac{1}{2}(b-a) \right )\,.$


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08533-9
PII: S 0002-9939(06)08533-9
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
Article copyright: © Copyright 2006 by the authors