Estimations Involving a Modulus of Continuity for a Generalization of Korovkin’s Operators

Abstract

In 1959 Korovkin [4] studied operators K_n: C[a,b] → C(a,b) of the form

$${K_n}(f;x) = \frac{1}{{{I_n}}}\int\limits_a^b {f(t)} {\beta ^n}(t - x)dt\quad \quad (n = 1,2,...), $$

In general this approximation does not take place at x = a and at x = b. Concerning the speed with which the approximation on (a,b) takes place Bojanic and Shisha [1] published in 1973 an investigation involving the modulus of continuity ω (δ) of f on [a,b]. However, they imposed on β rather severe restrictions, viz. β should not only satisfy Korovkin’s conditions but it should also be even on [-γ, γ] and monotonically decreasing on [0_, γ]. In addition they assumed that there exist two constants α > 0, c > 0 such that

$$ \mathop {\lim }\limits_{t \downarrow 0} \frac{{1 - \beta (t)}}{{{t^\alpha }}} = c. $$

.