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 Hermite interpolate S(rho)f(t) that involves only differences. Further, explicit error bounds are offered in the form of the inequality vertical bar vertical bar f - S(rho)f vertical bar vertical bar <= c(j) max (t is an element of N[a, b+2-j]) vertical bar Delta(j) f(t)vertical bar, 2 <= j <= 6 where the constants c(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.