TY  - BOOK
AU  - Totik,Vilmos
TI  - Polynomial approximation on polytopes
T2  - Memoirs of the American Mathematical Society
SN  - 9781470416669 (pbk. : acidfree paper)
U1  - 510 23
PY  - 2014///
CY  - Providence :  
PB  - American Mathematical Society,   
KW  - Polytopes
KW  - Orthogonal polynomials
KW  - Riemannian Geometry
N1  - "Volume 232, number 1091 (third of 6 numbers), November 2014."; Includes bibliographical references (pages 111-112); Part 1. The continuous case:
1. The result--
2. Outline of the proof--
3. Fast decreasing polynomials--
4. Approximation on simple polynomials--
5. Polynomial approximants on rhombi--
6. Pyramids and local moduli on them--
7. Local approximation on the sets Ka--
8. Global approximation of F=Fn on S1/32 excluding a neighborhood of the apex--
9. Global approximation of f on S1/64--
10. Completion of the froof of theorem 1.1--
11. Approximation in Rd--
12. A K-functional and the equivalence theorem--

Part 2. The Lp-case:
13. The Lp result--
14. Proof of the Lp result--
15. The dyadic decomposition--
16. Some properties of Lp moduli of smoothness--
17. Local Lp moduli of smoothness--
18. Local approximation--
19. Global Lp approximation excluding a neighborhood of the apex--
20. Strong direct and converse inequalities--
21. The K-functional in Lp and the equivalence theorem--
Acknowledgment--
Bibliography
N2  - Polynomial approximation on convex polytopes in d is considered in uniform and Lp-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the Lp -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
ER  -