L p-approximation by Kantorovič operators

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

“…The saturation case α = 1 in L p (p> 1) was settled by Riemenschneider [13] and Maier [12]: For the nonoptimal case 0 < α < 1 they stated a conjecture…”

The L p -saturation for linear combination of Bernstein-Kantorovich operators

“…On the other hand, as shown in [4,5,6], the saturation class for p = 1 and r = 1 is not the set of f satisfying ω 2 ϕ (f, t) 1 = O(t 2 ). Thus, from Theorems 1.1 and 1.2 we conclude that there exists…”

“…The saturation problem, that is, how to give an equivalence relation like (1.2) with α = 2r, is still open. The only known results of this kind are for r = 1 (see [4,5,6]). One may therefore conjecture that for r > 1, (1.2) should hold with α = 2r.…”

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

“…From Theorems 2 and 3 and the results of Maier [12] and Riemenschneider [13] As mentioned in [7], L~-saturation classes of both operators do not coincide. 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.…”

“…Converse Theorems for Approximation 389 Now we apply the standard technique for determining the saturation class of L,d" (see, e:g., [12]). …”