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.
Let x<sub>1</sub>, x<sub>2</sub>, . . ., x<sub>n-1</sub> be an increasing sequence of real numbers. Let g(x) be a piecewise polynomial defined by
where g<sub>1</sub>(x), . . ., g<sub>n</sub>(x) are polynomials of degree 3. Let h > 0. If
then g(x) is called a discrete cubic spline.
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
This states that the central differences are continuous at x<sub>i</sub>.
Let x<sub>1</sub> = 1 and x<sub>2</sub> = 2 so that n = 3. The following function defines a discrete cubic spline:
Let x<sub>0</sub> < x<sub>1</sub> and x<sub>n</sub> > x<sub>n-1</sub> and f(x) be a function defined in the closed interval [x<sub>0</sub> - h, x<sub>n</sub> + 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<sub>0</sub> - h, x<sub>n</sub> + h]. This interpolant agrees with the values of f(x) at x<sub>0</sub>, x<sub>1</sub>, . . ., x<sub>n</sub>.