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Polynomial approximation on polytopes / Vilmos Totik.

By: Material type: TextTextSeries: Memoirs of the American Mathematical Society ; v 232, no 1091.Publication details: Providence : American Mathematical Society, c2014.Description: v, 112 p. : illustrations ; 25 cmISBN:
  • 9781470416669 (pbk. : acidfree paper)
Subject(s): DDC classification:
  • 510 23 Am512
Contents:
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.
Summary: 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.
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"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.

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.

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