Abstract. There are several equivalence assertions known for the approximation of functions, continuous on an interval, by sequences of positive linear operators (e.g., operators of exponential type). Due to the improved rate of convergence near the endpoints of the interval under consideration, all these results indeed possess a pointwise (weighted) structure. In particular in connection with characterizations via the Ditzian--Totik modul, the present paper contributes to the question whether the individual direct and inverse estimates are sharp on large point sets. In fact, with the aid of an appropriate quantitative resonance principle and Borel--Cantelli-type result, the existence of counterexamples is established, simultaneously delivering the sharpness of the Jackson as well as of the Berenstein estimate on sets of full measure. To apply the general arguments mentioned, it is essential to determine relevant resonance elements. To this end, a unifying approach is suggested, based on the construction of a suitable matrix of knots.

AMS Subject Classification (1991): 41A25, 41A36

Received September 30, 1994. (Registered under 5640/2009.)