New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
 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 Not yet published.Expected publication date is October 24, 2014. 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. Table of Contents 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 The dyadic decomposition Some properties of $$L^p$$ moduli of smoothness Local $$L^p$$ moduli of smoothness Local approximation Global $$L^p$$ approximation excluding a neighborhood of the apex Strong direct and converse inequalities The $$K$$-functional in $$L^p$$ and the equivalence theorem Bibliography