Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
DE
EN
Books
Journals
Events
Topic Page
Marketing
Start single access now
Access for companies
12th International Engine Congress 2025 | 25. + 26. February 2025

Meeting Place for the Powertrain and Sustainable Fuels Community

The congress will examine the future role of the internal combustion engine and its non-fossil fuels from a global perspective with leading experts. Be part of it! More information and booking options are available on our website.
EXTENDED SEARCH
Log in
Top
Foundations of Computational Mathematics
Published in:

23-01-2023

Optimal Polynomial Meshes Exist on any Multivariate Convex Domain

Authors: Feng Dai, Andriy Prymak

Published in: Foundations of Computational Mathematics | Issue 3/2024

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We show that optimal polynomial meshes exist for every convex body in \({\mathbb {R}}^d\), confirming a conjecture by A. Kroó.
previous article On the Sample Complexity of Stabilizing Linear Dynamical Systems from Data
next article Nodal Auxiliary Space Preconditioning for the Surface de Rham Complex

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Footnotes
1
There appears to be no consistency in the literature for the names of the maximal volume-inscribed ellipsoid and the minimal volume-circumscribed ellipsoid; either one may be referred to as John’s or Löwner-John’s or Löwner’s ellipsoid. According to Busemann, Löwner discovered the uniqueness of the minimal volume ellipsoid, but this was never published. John established a characterization for these ellipsoids which implied uniqueness and other properties. An interested reader is referred to the survey [10].
 
Literature
1.
Bloom, T., Bos, L. P., Calvi, J.-P., Levenberg, N., Polynomial interpolation and approximation in \(\mathbb{C}^d\), Ann. Polon. Math., 106, 2012, 53–81,MathSciNetCrossRef
2.
Bos, L., Calvi, J.-P., Levenberg, N., Sommariva, A., Vianello, M., Geometric weakly admissible meshes, discrete least squares approximations and approximate Fekete points, Math. Comp., 80, 2011, 275, 1623–1638,MathSciNetCrossRef
3.
4.
Boyd, Stephen, Vandenberghe, Lieven, Convex optimization, Cambridge University Press, Cambridge, 2004,
5.
Dai, F., Prymak, A., Temlyakov, V. N., Tikhonov, S. Yu., Integral norm discretization and related problems, Russian, with Russian summary, Uspekhi Mat. Nauk, 74, 2019, 4(448), 3–58, Russian Math. Surveys, 74, 2019, 4, 579–630, ,
6.
De Marchi, Stefano, Marchioro, Martina, Sommariva, Alvise, Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder, Appl. Math. Comput., 218, 2012, 21, 10617–10629,MathSciNetCrossRef
7.
DeVore, Ronald A., Lorentz, George G., Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, 1993, x+449,
8.
9.
10.
Henk, Martin, Löwner-John ellipsoids, Doc. Math., 2012, Extra vol.: Optimization stories, 95–106,
11.
12.
Kroó, András, Bernstein type inequalities on star-like domains in \(\mathbb{R}^d\) with application to norming sets, Bull. Math. Sci., 3, 2013, 3, 349–361,MathSciNetCrossRef
13.
Kroó, András, Christoffel functions on convex and starlike domains in \(\mathbb{R}^d\), J. Math. Anal. Appl., 421, 2015, 1, 718–729,MathSciNetCrossRef
14.
15.
Kroó, András, On the existence of optimal meshes in every convex domain on the plane, J. Approx. Theory, 238, 2019, 26–37,MathSciNetCrossRef
16.
Jetter, Kurt, Stöckler, Joachim, Ward, Joseph D., Error estimates for scattered data interpolation on spheres, Math. Comp., 68, 1999, 226, 733–747,MathSciNetCrossRef
17.
Mastroianni, G., Totik, V., Weighted polynomial inequalities with doubling and \(A_\infty \) weights, Constr. Approx., 16, 2000, 1, 37–71,MathSciNetCrossRef
18.
Piazzon, Federico, Optimal polynomial admissible meshes on some classes of compact subsets of \(\mathbb{R}^d\), J. Approx. Theory, 207, 2016, 241–264,MathSciNetCrossRef
19.
Prymak, A., Geometric computation of Christoffel functions on planar convex domains, J. Approx. Theory, 268, 2021, Paper No. 105603, 13,
20.
Roberts, A. Wayne, Varberg, Dale E., Convex functions, Pure and Applied Mathematics, Vol. 57, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973, xx+300,
21.
Schneider, Rolf, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, 151, Second expanded edition, Cambridge University Press, Cambridge, 2014, xxii+736,
Metadata
Title
Optimal Polynomial Meshes Exist on any Multivariate Convex Domain
Authors
Feng Dai
Andriy Prymak
Publication date
23-01-2023
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 3/2024
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-023-09606-x

Other articles of this Issue 3/2024

A Stirling-Type Formula for the Distribution of the Length of Longest Increasing Subsequences

  • Open Access

Correction to: Conormal Spaces and Whitney Stratifications

  • Open Access
  • Correction

Subexponential-Time Algorithms for Sparse PCA

Nodal Auxiliary Space Preconditioning for the Surface de Rham Complex

A New Approach to Handle Curved Meshes in the Hybrid High-Order Method

  • Open Access

On the Sample Complexity of Stabilizing Linear Dynamical Systems from Data

Premium Partner

    Image Credits