An alternative variable projection formulation for small-size separable models

We consider several small-size Prony-like separable least squares interpolation problems in exponential, polynomial and mixed data fitting schemes, for which the variable projection objective function can easily be written down in closed form. An alternative formulation of variable projection is then applied to the closed form expression, yielding explicit formulas for the unknown parameters. The technique is especially useful in various exponential type spline generalizations where the free frequency parameter is otherwise mostly determined by trial and error.