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Polynomial Approximation on Polytopes
Vilmos Totik, Bolyai Institute, University of Szeged, Hungary
Memoirs of the American Mathematical Society
2014; 112 pp; softcover
Volume: 232
ISBN-10: 1-4704-1666-2
ISBN-13: 978-1-4704-1666-9
List Price: US$75
Individual Members: US$45
Institutional Members: US$60
Order Code: MEMO/232/1091
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Not yet published.
Expected publication date is October 24, 2014.

Polynomial approximation on convex polytopes in \(\mathbf{R}^d\) is considered in uniform and \(L^p\)-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the \(L^p\)-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate \(K\)-functional follows as a consequence.

The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.

Table of Contents

Part 1. The continuous case
  • The result
  • Outline of the proof
  • Fast decreasing polynomials
  • Approximation on simple polytopes
  • Polynomial approximants on rhombi
  • Pyramids and local moduli on them
  • Local approximation on the sets \(K_a\)
  • Global approximation of \(F=F_n\) on \(S_{1/32}\) excluding a neighborhood of the apex
  • Global approximation of \(f\) on \(S_{1/64}\)
  • Completion of the proof of Theorem 1.1
  • Approximation in \(\mathbf{R}^d\)
  • A \(K\)-functional and the equivalence theorem
Part 2. The \(L^p\)-case
  • The \(L^p\) result
  • Proof of the \(L^p\) result
  • The dyadic decomposition
  • Some properties of \(L^p\) moduli of smoothness
  • Local \(L^p\) moduli of smoothness
  • Local approximation
  • Global \(L^p\) approximation excluding a neighborhood of the apex
  • Strong direct and converse inequalities
  • The \(K\)-functional in \(L^p\) and the equivalence theorem
  • Bibliography
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