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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.

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On the norms of discrete analogues of convolution operators
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by Oleg Kovrizhkin PDF
Proc. Amer. Math. Soc. 140 (2012), 1349-1352 Request permission

Abstract:

We consider a discrete analogue of convolution operator $T(f) = K*f$ from $L^p({\mathbb {R}}^d)$ to $L^p({\mathbb {R}}^d)$: $T_{dis}(g) = K_{dis}* g$ from $\ell ^p(\mathbb {Z}^d)$ to $\ell ^p(\mathbb {Z}^d)$ where $K_{dis} = K \vert _{\mathbb {Z}^d}$ and $\hat K$ is supported in the fundamental cube. We show that the estimate $\|T_{dis}\|_p \le C^d \|T\|_p$ with $C > 1$ cannot be improved for a certain range of $p$.
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Additional Information
  • Oleg Kovrizhkin
  • Email: olegk@alum.mit.edu
  • Received by editor(s): December 16, 2010
  • Received by editor(s) in revised form: January 4, 2011
  • Published electronically: August 4, 2011
  • Additional Notes: This research was partially supported by NSF grant DMS 0201099
  • Communicated by: Michael T. Lacey
  • © Copyright 2011 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1349-1352
  • MSC (2010): Primary 42A99, 42B99
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11043-8
  • MathSciNet review: 2869118