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 x1, x2,..., xn-1 be an increasing sequence of real numbers. Let g be a piecewise polynomial defined by
where g1,..., gn are polynomials of degree 3. Let h > 0. If
then g 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 are defined as follows:
The conditions defining a discrete cubic spline are also equivalent to
This states that the central differences are continuous at xi.

Example

Let x1 = 1 and x2 = 2 so that n = 3. The following function defines a discrete cubic spline:

Discrete cubic spline interpolant

Let x0 < x1 and xn > xn-1 and f be a function defined in the closed interval . Then there is a unique cubic discrete spline g satisfying the following conditions:
This unique discrete cubic spline is the discrete spline interpolant to f in the interval . This interpolant agrees with the values of f at x0, x1,..., xn.

Applications