Memoirs of the American Mathematical Society 2014; 112 pp; softcover Volume: 232 ISBN10: 1470416662 ISBN13: 9781470416669 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, socalled 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
