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Transactions of the American Mathematical Society

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On the growth of Lebesgue constants for convex polyhedra


Authors: Yurii Kolomoitsev and Tetiana Lomako
Journal: Trans. Amer. Math. Soc. 370 (2018), 6909-6932
MSC (2010): Primary 42B05, 42B15, 42B08
DOI: https://doi.org/10.1090/tran/7225
Published electronically: April 4, 2018
MathSciNet review: 3841836
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Abstract: In this paper, new estimates of the Lebesgue constant \begin{equation*} \mathcal {L}(W)=\frac 1{(2\pi )^d}\int _{{\Bbb T}^d}\bigg |\sum _{\mathbfit {k}\in W\cap {\Bbb Z}^d} e^{i(\mathbfit {k}, \mathbfit {x})}\bigg | \textrm {d}\mathbfit {x} \end{equation*} for convex polyhedra $W\subset {\Bbb R}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times \dots \times [0,m_d]\subset W\subset [0,n_1]\times \dots \times [0,n_d]$, then \begin{equation*} c(d)\prod _{j=1}^d \log (m_j+1)\le \mathcal {L}(W)\le C(d)s\prod _{j=1}^d \log (n_j+1), \end{equation*} where $s$ is a size of the triangulation of $W$.


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Additional Information

Yurii Kolomoitsev
Affiliation: Institute of Applied Mathematics and Mechanics of NAS of Ukraine, General Batyuk Str. 19, Slov’yans’k, Donetsk region, Ukraine, 84100
Address at time of publication: Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany
MR Author ID: 790232
ORCID: 0000-0001-7613-4169
Email: kolomoitsev@math.uni-luebeck.de, kolomus1@mail.ru

Tetiana Lomako
Affiliation: Institute of Applied Mathematics and Mechanics of NAS of Ukraine, General Batyuk Str. 19, Slov’yans’k, Donetsk region, Ukraine, 84100
Address at time of publication: Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, 23562 Lübeck, Germany
MR Author ID: 893561
Email: tlomako@yandex.ru

Keywords: Lebesgue constants, Dirichlet kernel, convex polyhedra
Received by editor(s): August 24, 2016
Received by editor(s) in revised form: January 15, 2017
Published electronically: April 4, 2018
Additional Notes: This research was supported by H2020-MSCA-RISE-2014 Project number 645672 (AMMODIT: “Approximation Methods for Molecular Modelling and Diagnosis Tools”)
Article copyright: © Copyright 2018 American Mathematical Society