The paper is in essence a survey of categories having φ-weighted colimits for all the weights φ in some class Φ. We introduce the class Φ of Φ-flat weights which are those ψ for which ψ-colimits commute in the base V with limits having weights in Φ; and the class Φ− of Φ-atomic weights, which are those ψ for which ψ-limits commute in the base V with colimits having weights in Φ. We show that bo...

For a general class of exponential weights on the line and on (−1, 1), we study pointwise convergence of the derivatives of Lagrange interpolation. Our weights include even weights of smooth polynomial decay near ±∞ (Freud weights), even weights of faster than smooth polynomial decay near ±∞ (Erdős weights) and even weights which vanish strongly near ±1, for example Pollaczek type weights. 1991...