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Transactions of the American Mathematical Society

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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
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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.

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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

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.