Mixed interpolation methods with arbitrary nodes

Previous work on interpolation by linear combinations of the form aC(x) + bS(x) + ∑_i=0ⁿ⁻²α_ixⁱ, where C and S are given functions and the coefficients a, b, and {α_j} are determined by the interpolation conditions, was restricted to uniformly spaced interpolation nodes. Here we derive both Newtonian and Lagrangian formulae for the interpolant for arbitrarily chosen distinct nodes. In the Newtonian form the interpolating function is expressed as the sum of the interpolating polynomial based on the given nodes and two correction terms involving an auxiliary function for which a recurrence relation is obtained. Each canonical function for the Lagrangian form may be expressed as a product of the corresponding Lagrange polynomial and a function which depends on divided differences of C(x) and S(x).