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Polynomial approximation on polytopes
About this Title
Vilmos Totik, MTA-SZTE Analysis and Stochastics, Research Group, Bolyai Institute, University of Szeged, Szeged. Aradi v. tere 1, 6720, Hungary — and — Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, CMC342, Tampa, Florida 33620-5700
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: https://doi.org/10.1090/memo/1091
Published electronically: March 5, 2014
Keywords: Polynomials,
several variables,
approximation,
moduli of smoothness,
$K$-functionals,
strong inequalities
MSC: Primary 41A10, 41A17
Table of Contents
1. The continuous case
- 1. The result
- 2. Outline of the proof
- 3. Fast decreasing polynomials
- 4. Approximation on simple polytopes
- 5. Polynomial approximants on rhombi
- 6. Pyramids and local moduli on them
- 7. Local approximation on the sets $K_a$
- 8. Global approximation of $F=F_n$ on $S_{1/32}$ excluding a neighborhood of the apex
- 9. Global approximation of $f$ on $S_{1/64}$
- 10. Completion of the proof of Theorem
- 11. Approximation in ${\mathbf R}^d$
- 12. A $K$-functional and the equivalence theorem
2. The $L^p$-case
- 13. The $L^p$ result
- 14. Proof of the $L^p$ result
- 15. The dyadic decomposition
- 16. Some properties of $L^p$ moduli of smoothness
- 17. Local $L^p$ moduli of smoothness
- 18. Local approximation
- 19. Global $L^p$ approximation excluding a neighborhood of the apex
- 20. Strong direct and converse inequalities
- 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.- F. Dai, Z. Ditzian, and Hongwei Huang, Equivalence of measures of smoothness in $L_p(S^{d-1})$, $1<p<\infty$, Studia Math. 196 (2010), no. 2, 179–205. MR 2570335, DOI 10.4064/sm196-2-5
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