| eBook ISBN: | 978-1-4704-1894-6 |
| Product Code: | MEMO/232/1091.E |
| List Price: | $75.00 |
| MAA Member Price: | $67.50 |
| AMS Member Price: | $45.00 |
| eBook ISBN: | 978-1-4704-1894-6 |
| Product Code: | MEMO/232/1091.E |
| List Price: | $75.00 |
| MAA Member Price: | $67.50 |
| AMS Member Price: | $45.00 |
-
Book DetailsMemoirs of the American Mathematical SocietyVolume: 232; 2014; 112 ppMSC: Primary 41
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
-
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
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Requests
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.
-
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
