Prof. Avraham Melkman

The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.