Polynomial Approximation on Polytopes.

Cover -- Title page -- Part \ 1 . The continuous case -- Chapter 1. The result -- Chapter 2. Outline of the proof -- Chapter 3. Fast decreasing polynomials -- Chapter 4. Approximation on simple polytopes -- Chapter 5. Polynomial approximants on rhombi -- Chapter 6. Pyramids and local moduli on them -- Chapter 7. Local approximation on the sets ₐ -- Chapter 8. Global approximation of = _{ } on _{1/32} excluding a neighborhood of the apex -- Chapter 9. Global approximation of on _{1/64} -- Chapter 10. Completion of the proof of Theorem 1.1 -- Chapter 11. Approximation in \R^{ } -- Chapter 12. A -functional and the equivalence theorem -- Part \ 2 . The ^{ }-case -- Chapter 13. The ^{ } result -- Chapter 14. Proof of the ^{ } result -- Chapter 15. The dyadic decomposition -- Chapter 16. Some properties of ^{ } moduli of smoothness -- Chapter 17. Local ^{ } moduli of smoothness -- Chapter 18. Local approximation -- Chapter 19. Global ^{ } approximation excluding a neighborhood of the apex -- Chapter 20. Strong direct and converse inequalities -- Chapter 21. The -functional in ^{ } and the equivalence theorem -- Acknowledgement -- Bibliography -- Back Cover.

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.

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