Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Nikolskii-type inequalities for shift invariant function spaces
HTML articles powered by AMS MathViewer

by Peter Borwein and Tamás Erdélyi PDF
Proc. Amer. Math. Soc. 134 (2006), 3243-3246

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 \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 $\delta \in \left (0,\frac {1}{2}(b-a) \right ) .$

References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A17
  • Retrieve articles in all journals with MSC (2000): 41A17
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
  • Received by editor(s): May 17, 2005
  • Published electronically: June 6, 2006
  • Communicated by: David Preiss
  • © Copyright 2006 by the authors
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3243-3246
  • MSC (2000): Primary 41A17
  • DOI: https://doi.org/10.1090/S0002-9939-06-08533-9
  • MathSciNet review: 2231907