Electronic ISBN:  9781470418946 
Product Code:  MEMO/232/1091.E 
112 pp 
List Price:  $75.00 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 232; 2014MSC: 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, socalled 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


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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, socalled 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