Approximation by discrete spline interpolation

For a function f(t) defined on the discrete interval N[a, b + 2] = {a, a + 1, ..., b + 2}, we develop a class of quintic discrete spline interpolate S_ρ f(t) that involves only differences. Further, explicit error bounds are offered in the form of the inequality ||f-H_ρ f||≤d_j max/tϵN[a, b+2-j] |Δ^j f(t)|, 2≤j≤6 where the constants d_j, 2 ≤ j ≤ 6 are explicitly provided. Three numerical examples are presented to illustrate the actual construction of the discrete spline interpolates, the actual errors are also computed to compare with the error bounds obtained.