Polynomial approximation on convex polytopes in Rd is considered in uniform
and Lp-norms. For an appropriate modulus of smoothness matching direct and
converse estimates are proven. In the Lp-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 1980’s when some of the present findings were
established for special, so called simple polytopes.
Received by the editor February 21, 2012.
Article electronically published on March 5, 2014.
2010 Mathematics Subject Classification. Primary 41A10, 41A17.
Key words and phrases. Polynomials, several variables, approximation, moduli of smooth-
ness, K-functionals, strong inequalities.
Aﬃliation at time of publication: MTA-SZTE Analysis and Stochastics, Research Group,
Bolyai Institute, University of Szeged, Szeged. Aradi v. tere 1, 6720, Hungary; and Department
of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, CMC342, Tampa,
Florida 33620-5700, email: firstname.lastname@example.org.
2014 American Mathematical Society