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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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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

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