**Memoirs of the American Mathematical Society**

2014;
112 pp;
Softcover

MSC: Primary 41;

Print ISBN: 978-1-4704-1666-9

Product Code: MEMO/232/1091

List Price: $75.00

Individual Member Price: $45.00

**Electronic ISBN: 978-1-4704-1894-6
Product Code: MEMO/232/1091.E**

List Price: $75.00

Individual Member Price: $45.00

# Polynomial Approximation on Polytopes

Share this page
*Vilmos Totik*

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

# Table of Contents

## Polynomial Approximation on Polytopes

- Cover Cover11 free
- Title page i2 free
- Part \ 1 . The continuous case 18 free
- Chapter 1. The result 310
- Chapter 2. Outline of the proof 714
- Chapter 3. Fast decreasing polynomials 1118
- Chapter 4. Approximation on simple polytopes 1522
- Chapter 5. Polynomial approximants on rhombi 2330
- Chapter 6. Pyramids and local moduli on them 2734
- Chapter 7. Local approximation on the sets 𝐾ₐ 3542
- Chapter 8. Global approximation of 𝐹=𝐹_{𝑛} on 𝑆_{1/32} excluding a neighborhood of the apex 3744
- Chapter 9. Global approximation of 𝑓 on 𝑆_{1/64} 3946
- Chapter 10. Completion of the proof of Theorem 1.1 4552
- Chapter 11. Approximation in \R^{𝑑} 4956
- Chapter 12. A 𝐾-functional and the equivalence theorem 5360

- Part \ 2 . The 𝐿^{𝑝}-case 5562
- Chapter 13. The 𝐿^{𝑝} result 5764
- Chapter 14. Proof of the 𝐿^{𝑝} result 6168
- Chapter 15. The dyadic decomposition 6774
- Chapter 16. Some properties of 𝐿^{𝑝} moduli of smoothness 7582
- Chapter 17. Local 𝐿^{𝑝} moduli of smoothness 8390
- Chapter 18. Local approximation 95102
- Chapter 19. Global 𝐿^{𝑝} approximation excluding a neighborhood of the apex 99106
- Chapter 20. Strong direct and converse inequalities 103110
- Chapter 21. The 𝐾-functional in 𝐿^{𝑝} and the equivalence theorem 109116
- Bibliography 111118

- Back Cover Back Cover1124