Sums of Weighted Differentiation Composition Operators

We solve an interpolation problem in Ap involving specifying a set of (possibly not distinct) n points, where the kth derivative at the kth point is up to a constant as large as possible for functions of unit norm. The solution obtained has norm bounded by a constant independent of the points chosen. As a direct application, we obtain a characterization of the order-boundedness of a sum of products of weighted composition and differentiation operators acting between weighted Bergman spaces. We also characterize the compactness of such operators that map a weighted Bergman space into the space of bounded analytic functions.