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memo_has_moved_text();Polynomial approximation on polytopes

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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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 $\mathbf {R}^d$ 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 1980's when some of the present findings were established for special, so called simple polytopes.