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.
[1] Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences.
., xn-1 be an increasing sequence of real numbers.
., gn(x) are polynomials of degree 3.
If then g(x) is called a discrete cubic spline.
[1] The conditions defining a discrete cubic spline are equivalent to the following: 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[1] This states that the central differences
The following function defines a discrete cubic spline:[1]
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 [x0 - h, xn + h].
This interpolant agrees with the values of f(x) at x0, x1, .