POINTWISE AND UNIFORM ESTIMATES FOR CONVEX APPROXIMATION OF FUNCTIONS BY ALGEBRAIC POLYNOMIALS

Let DELTA(q) be the set of functions f for which the qth difference is non-negative on the interval [-1, 1], P(n) is the set of algebraic polynomials of degree not exceeding n, tau(k)(f, delta)p is the averaged Sendov-Popov modulus of smoothness in the L(p)[-1, 1] metric for 1 less-than-or-equal-to p less-than-or-equal-to infinity, omega(k)(f, delta) and omega(phi)f(f, delta), phi(x): = square-root 1 - x2, are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a function f is-an-element-of C[-1, 1] and DELTA2 we construct a polynomial p(n) is-an-element-of P(n) and DELTA2 such that \f(x) - p(n)(x)\ less-than-or-equal-to Comega3(f, n-1 square-root 1 - x2 + n-2), x is-an-element-of [-1, 1]; \\f - p(n)\\infinity less-than-or-equal-to Comega(phi)3(f, n-1); \\f - P(n)\\p less-than-or-equal-to Ctau3(f, n-1)p. As a consequence, for a function f is-an-element-of C2[-1, 1] and DELTA3 a polynomial p(n)* is-an-element-of P(n) and DELTA3 exists such that \\f - p(n)*\\infinity less-than-or-equal-to Cn-1omega2(f', n-1), where n greater-than-or-equal-to 2 and C is an absolute constant.