Contents
Part 1. The continuous case 1
Chapter 1. The result 3
Chapter 2. Outline of the proof 7
Chapter 3. Fast decreasing polynomials 11
Chapter 4. Approximation on simple polytopes 15
Chapter 5. Polynomial approximants on rhombi 23
Chapter 6. Pyramids and local moduli on them 27
Chapter 7. Local approximation on the sets Ka 35
Chapter 8. Global approximation of F = Fn on S1/32 excluding a
neighborhood of the apex 37
Chapter 9. Global approximation of f on S1/64 39
Chapter 10. Completion of the proof of Theorem 1.1 45
Chapter 11. Approximation in Rd 49
Chapter 12. A K-functional and the equivalence theorem 53
Part 2. The
Lp-case
55
Chapter 13. The
Lp
result 57
Chapter 14. Proof of the
Lp
result 61
Chapter 15. The dyadic decomposition 67
Chapter 16. Some properties of
Lp
moduli of smoothness 75
Chapter 17. Local Lp moduli of smoothness 83
Chapter 18. Local approximation 95
Chapter 19. Global Lp approximation excluding a neighborhood of the apex 99
Chapter 20. Strong direct and converse inequalities 103
Chapter 21. The K-functional in
Lp
and the equivalence theorem 109
iii
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