Forward interpolation, as discussed here, refers to the problem of interpolating data points revealed, one at a time, in a sequential manner. As such, forward spline interpolation is an inherently unstable process. In this paper we study forward interpolation using a class of "generalized splines": piecewise polynomials of odd degree 2m-1, continuous to order m-1, where m >= 2 is an integer. We show that the problem can be interpreted as a state estimation problem for a discrete linear system and show how successful, stable procedures can be obtained by applying optimal state estimation techniques of "LQG" type long familiar in the theory of such systems. A number of examples are presented.