POLYNOMIAL AND RATIONAL APPROXIMATION OF FUNCTIONS OF SEVERAL VARIABLES WITH CONVEX DERIVATIVES IN THE LP-METRIC (0-LESS-THAN-P-LESS-THAN-OR-EQUAL-TO-INFINITY)

Let Conv(n)((l))(G) be the set of all functions f such that for every n-dimensional unit vector e the lth derivative in the direction of e, D-(l)(e)f, is continuous on a convex bounded domain G subset of R(n) (n greater than or equal to 2) and convex (upwards or downwards) on the nonempty intersection of every line L subset of R(n) with the domain G, and let M((l))(f, G) := sup{parallel to D-(l)(e)f parallel to(C(G)) : e is an element of R(n), parallel to e parallel to = 1} < infinity. Sharp, in the sense of order of smallness, estimates of best simultaneous polynomial approximations of the functions f is an element of Conv(n)((l))(G), for which D(l)(e)f is an element of Lip(K) 1 for every e, and their derivatives in the metrics of L(p)(G) (0 < p less than or equal to infinity are obtained. It is proved that the corresponding parts of these estimates are preserved for best rational approximations, on any n-dimensional parallelepiped Q, of functions f is an element of Conv(n)((l))(Q) in the metrics of Lp(Q) (0 < p < infinity) and it is shown that they are sharp in the sense of order of smallness for 0 < p less than or equal to 1.