Algebraic-trigonometric mixed Hermite interpolation and the error estimation

The function f(x) is approximated by interpolation function fn(x) = a cos kx+b sin kx+qq[cihi(x) +c over-bar ih over-bar i(x)] such that f(xi) = fn(xi) and f prime (xi) = f prime n(xi) (i = 0, 1, ..., n). fn(x) is proved to be unique and is tending to Hermite interpolation polynomials H2n+1(f, x) as the parameter k-&gt0. The error term is also discussed, and as an application, a class of extended linear multistep methods of Adams' type for equidistant point are established.