Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Growth of $ L^{p}$ Lebesgue constants for convex polyhedra and other regions

Authors: J. Marshall Ash and Laura De Carli
Journal: Trans. Amer. Math. Soc. 361 (2009), 4215-4232
MSC (2000): Primary 42B15, 42A05; Secondary 42B08, 42A45
Published electronically: March 4, 2009
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any convex polyhedron $ W$ in $ \mathbb{R}^{m}$, $ p\in\left(1,\infty \right) $, and $ N\geq1$, there are constants $ \gamma_{1}\left(W,p,m\right) $ and $ \gamma_{2}\left(W,p,m\right) $ such that

$\displaystyle \gamma_{1}N^{m\left(p-1\right) }\leq\int_{\mathbb{T}^{m}}\left\ve... ...e\left(k\cdot x\right) \right\vert ^{p}dx\leq\gamma _{2}N^{m\left(p-1\right)}. $

Similar results hold for more general regions. These results are various special cases of the inequalities

$\displaystyle \gamma_{1}N^{m\left(p-1\right) }\leq\int_{\mathbb{T}^{m}}\left\ve... ...NB}e\left(k\cdot x\right) \right\vert ^{p}dx\leq\gamma_{2} \phi\left(N\right), $

where $ \phi\left(N\right)=N^{p\left(m-1\right) /2}$ when $ p\in\left( 1,\frac{2m}{m+1}\right)$, $ \phi\left(N\right)=N^{p\left(m-1\right) /2}\log$ $ N$ when $ p=\frac{2m}{m+1}$, and $ \phi\left(N\right)=N^{m\left( p-1\right) }$ when $ p>\frac{2m}{m+1}$, where $ B$ is a bounded subset of $ \mathbb{R}^{m}$ with non-empty interior.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B15, 42A05, 42B08, 42A45

Retrieve articles in all journals with MSC (2000): 42B15, 42A05, 42B08, 42A45

Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614

Laura De Carli
Affiliation: Department of Mathematics, Florida International University, University Park, Miami, Florida 33199

PII: S 0002-9947(09)04627-3
Keywords: Lebesgue constant, Dirichlet kernel, kernels for convex sets, kernels for polyhedra
Received by editor(s): January 16, 2007
Received by editor(s) in revised form: August 3, 2007
Published electronically: March 4, 2009
Additional Notes: The first author’s research was partially supported by a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia