L P ( ρ > 1)-APPROXIMATION BY KANTOROVICH POLYNOMIALS

Approximation Strategy by Positive Linear Operators

“…which has been studied most widely among the positive linear operators of the form (2) (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]). Interested readers could also refer to the related papers for the other similar operators.…”

Some properties of a new kind of modulus of smoothness

“…(iii) If q and 9 2 are two functions with the above properties, then Kçv2 implies K1w(f, with a K1 independent of f € B and ô > 0, (iv) If w(f, 5) o(b2) /or a sequence ô -0 then / is linear.…”

“…(ii) There is a constant K for which w(f, ,ô) :5, K22w (5) is satisfied for all / € B, 2 > I and 6 > 0.…”

Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1<p<∞)

“…Let us compare the rate of approximation by Bernstein polynomials with the rate of approximation by Kantorovich polynomials K,f (See Section 2) and the best approximation of f in Lp[0, 1] by algebraic polynomials of nth degree E,(f) r As a consequence of Theorem 1, Corollary 4.1 in [6] and Totik's results in [14] (see also Lemmas 7 and 8 below) we have When a <-I/p, statements (b) and (c) in Corollary 2 are also equivalent [15, p. 239], but we cannot expect an equivalence of (a) and (b) for each a < 1 because of the discrete nature of Bernstein polynomials.…”