Abstract: | Let ga(t) and gb(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {Ln} be a sequence of linear positive operators, each with domain containing 1, t, ga(t), and gb(t). If Ln(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, ga(t), gb(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(ga(t)) (t → a+) and ƒ(t) = O(gb(t)) (t → b−). We estimate the convergence rate of Lnƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′. |