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Polynomial Approximation on Polytopes
Vilmos Totik, Bolyai Institute, University of Szeged, Hungary
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Memoirs of the American Mathematical Society
2014; 112 pp; softcover
Volume: 232
ISBN-10: 1-4704-1666-2
ISBN-13: 978-1-4704-1666-9
List Price: US$75 Individual Members: US$45
Institutional Members: US\$60
Order Code: MEMO/232/1091

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.

Part 1. The continuous case
• The result
• Outline of the proof
• Fast decreasing polynomials
• Approximation on simple polytopes
• Polynomial approximants on rhombi
• Pyramids and local moduli on them
• Local approximation on the sets $$K_a$$
• Global approximation of $$F=F_n$$ on $$S_{1/32}$$ excluding a neighborhood of the apex
• Global approximation of $$f$$ on $$S_{1/64}$$
• Completion of the proof of Theorem 1.1
• Approximation in $$\mathbf{R}^d$$
• A $$K$$-functional and the equivalence theorem
Part 2. The $$L^p$$-case
• The $$L^p$$ result
• Proof of the $$L^p$$ result
• Some properties of $$L^p$$ moduli of smoothness
• Local $$L^p$$ moduli of smoothness
• Global $$L^p$$ approximation excluding a neighborhood of the apex
• The $$K$$-functional in $$L^p$$ and the equivalence theorem