Approximation by discrete hermite interpolation

For a function f(t) defined on the discrete interval N[a, b + 2] = {a, a + 1, ..., 6 + 2}, we develop a class of quintic discrete Hermite interpolate H_ρ f(t) that involves only differences. Further, explicit error bounds are offered in the form of the inequality ||f-H_ρ f||≤c_j max/tϵN[a, b+2-j] | Δ^j f(t)|, 2≤j≤6 where the constants C_j, 2 ≤ 6 are explicitly provided. We also establish the two-variable Hermite interpolate as well as the related error analysis for a function f(t, u) defined on N[a, b+2] × N[c, d + 2]. Four numerical examples are presented to illustrate the actual construction of the discrete Hermite interpolates, the actual errors are also computed to compare with the error bounds obtained.