Discrete spline interpolation
In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous. Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain  minimization problems involving differences. Discrete cubic splines. Let "x", "x", . . ., "x" be an increasing sequence of real numbers. Let "g"("x") be a piecewise polynomial defined by where "g"("x"), . . ., "g"("x") are polynomials of degree 3. Let "h" > 0. If then "g"("x") is called a discrete cubic spline. Alternative formulation 1. The conditions defining a discrete cubic spline are equivalent to the following: Alternative formulation 2. The central differences of orders 0, 1, and 2 of a function "f"("x") are defined as follows: The conditions defining a discrete cubic spline are also equivalent to This states that the central differences formula_10  are continuous at "x". Example. Let "x" = 1 and "x" = 2 so that "n" = 3. The following function defines a discrete cubic spline: formula_11 Discrete cubic spline interpolant. Let "x" < "x" and "x" > "x" and "f"("x") be a function defined in the closed interval ["x" - h, "x" + h]. Then there is a unique cubic discrete spline "g"("x") satisfying the following conditions: This unique discrete cubic spline is the discrete spline interpolant to "f"("x") in the interval ["x" - h, "x" + h]. This interpolant agrees with the values of "f"("x") at "x", "x", . . ., "x". Discrete