On almost-Hermite-Fejér-interpolation: pointwise estimates

Abstract

We give a brief survey of the results obtained by numerous authors in so-called almost-Hermite-Fejér-interpolation and deal mainly with new quantitative assertions.These are based upon more general theorems for certain continuous linear operators which yield estimates involving different types of moduli of continuity.Our paper shows that in the case of almost-Hermite-Fejér-interpolation the underlying general technique can be used to treat three essentially different cases: sequences of positive operators, which converge uniformly for every continuous function on [−1, 1], sequences of non-positive operators doing the same, and sequences of operators which converge on proper subspaces of C[−1, 1] only.