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Polynomial approximation on polytopes


About this Title

Vilmos Totik

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1091
ISBNs: 978-1-4704-1666-9 (print); 978-1-4704-1894-6 (online)
DOI: http://dx.doi.org/10.1090/memo/1091
Published electronically: March 5, 2014
Keywords:Polynomials, several variables, approximation, moduli of smoothness, $K$-functionals

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Table of Contents


Chapters

  • Chapter 1. The result
  • Chapter 2. Outline of the proof
  • Chapter 3. Fast decreasing polynomials
  • Chapter 4. Approximation on simple polytopes
  • Chapter 5. Polynomial approximants on rhombi
  • Chapter 6. Pyramids and local moduli on them
  • Chapter 7. Local approximation on the sets $K_a$
  • Chapter 8. Global approximation of $F=F_n$ on $S_{1/32}$ excluding a neighborhood of the apex
  • Chapter 9. Global approximation of $f$ on $S_{1/64}$
  • Chapter 10. Completion of the proof of Theorem 1.1
  • Chapter 11. Approximation in ${\mathbf R}^d$
  • Chapter 12. A $K$-functional and the equivalence theorem
  • Chapter 13. The $L^p$ result
  • Chapter 14. Proof of the $L^p$ result
  • Chapter 15. The dyadic decomposition
  • Chapter 16. Some properties of $L^p$ moduli of smoothness
  • Chapter 17. Local $L^p$ moduli of smoothness
  • Chapter 18. Local approximation
  • Chapter 19. Global $L^p$ approximation excluding a neighborhood of the apex
  • Chapter 20. Strong direct and converse inequalities
  • Chapter 21. The $K$-functional in $L^p$ and the equivalence theorem

Abstract


Polynomial approximation on convex polytopes in is considered in uniform and -norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the -case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate -functional follows as a consequence. The results solve a problem that was left open since the mid 1980's when some of the present findings were established for special, so called simple polytopes.

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