TY  - BOOK
AU  - Totik,Vilmos
TI  - Polynomial Approximation on Polytopes
T2  - Memoirs of the American Mathematical Society Series
SN  - 9781470418946
AV  - QA691 .T68 2014 
U1  - 516/.158 
PY  - 2014///
CY  - Providence
PB  - American Mathematical Society
KW  - Geometry, Riemannian
KW  - Orthogonal polynomials
KW  - Polytopes
KW  - Electronic books
N1  - 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
N2  - 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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ER  -