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Polynomial Approximation on Polytopes

Vilmos Totik Bolyai Institute, University of Szeged, Hungary
Available Formats:
Electronic ISBN: 978-1-4704-1894-6
Product Code: MEMO/232/1091.E
112 pp
List Price: $75.00 MAA Member Price:$67.50
AMS Member Price: $45.00 Click above image for expanded view Polynomial Approximation on Polytopes Vilmos Totik Bolyai Institute, University of Szeged, Hungary Available Formats:  Electronic ISBN: 978-1-4704-1894-6 Product Code: MEMO/232/1091.E 112 pp  List Price:$75.00 MAA Member Price: $67.50 AMS Member Price:$45.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2322014
MSC: 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.

• 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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Volume: 2322014
MSC: 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.

• 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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