GENERAL RECURRENCE RELATION FOR DIVIDED DIFFERENCES AND THE GENERAL NEWTON-INTERPOLATION-ALGORITHM WITH APPLICATIONS TO TRIGONOMETRIC INTERPOLATION

In this note an ultimate generalization of Newton's classical interpolation formula is given. More precisely, we will establish the most general linear form of a Newton-like interpolation formula and a general recurrence relation for divided differences which are applicable whenever a function is to be interpolated by means of linear combinations of functions forming a Čebyšev-system such that at least one of its subsystems is again a Čebyšev-system. The theory is applied to trigonometric interpolation yielding a new algorithm which computes the interpolating trigonometric polynomial of smallest degree for any distribution of the knots by recurrence. A numerical example is given. © 1979 Springer-Verlag.