Our starting point is a basic problem in Hermite interpolation theory—namely, determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem in many cases if the number of points is small compared to the dimension of their linear span. This also allows us to establish results on the Hilbert function of ideals generated by powers of linear forms. The Verlinde formula determines such a Hilbert function in a specific instance. We complement this result and also determine the Castelnuovo–Mumford regularity of the corresponding ideals. As applications, we establish new instances of conjectures by Chudnovsky and by Demailly on the Waldschmidt constant. Moreover, we show that conjectures on the failure of the weak Lefschetz property by Harbourne, Schenck, and Seceleanu as well as by Migliore, Miró-Roig, and the first author are true asymptotically. The latter also relies on a new result for Eulerian numbers.