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(x) be a piecewise polynomial defined by



g(x)= \begin{cases} g_1(x) & x 0. If



(g_{i+1}-g_i)(x_i +jh)=0 \text{ for } j=-1,0,1 \text{ and } i=1,2,\ldots, n-1 $$

then g(x) is called a discrete cubic spline.

Alternative formulation 1
The conditions defining a discrete cubic spline are equivalent to the following:


 * $$ g_{i+1}(x_i-h) = g_i(x_i-h)$$


 * $$ g_{i+1}(x_i) = g_i(x_i)$$


 * $$ g_{i+1}(x_i+h) = g_i(x_i+h)$$

Alternative formulation 2
The central differences of orders 0, 1, and 2 of a function f(x) are defined as follows:


 * $$D^{(0)}f(x) = f(x) $$


 * $$D^{(1)}f(x)=\frac{f(x+h)-f(x-h)}{2h}$$


 * $$D^{(2)}f(x)=\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$$

The conditions defining a discrete cubic spline are also equivalent to


 * $$D^{(j)}g_{i+1}(x_i)=D^{(j)}g_i(x_i) \text{ for } j=0,1,2 \text{ and } i=1,2, \ldots, n-1.$$

This states that the central differences $$D^{(j)}g(x)$$ are continuous at xi.

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

$$ g(x) = \begin{cases} x^3 & x<1 \\ x^3 - 2(x-1)((x-1)^2-h^2) & 1\le x < 2\\ x^3 - 2(x-1)((x-1)^2-h^2)+(x-2)((x-2)^2-h^2) & x \ge 2 \end{cases} $$

Discrete cubic spline interpolant
Let x0 < x1 and xn > xn-1 and f(x) be a function defined in the closed interval [x0 - h, xn + h]. Then there is a unique cubic discrete spline g(x) satisfying the following conditions:


 * $$g(x_i) = f(x_i) \text{ for } i=0,1,\ldots, n.$$
 * $$D^{(1)}g_1(x_0) = D^{(1)}f(x_0).$$
 * $$D^{(1)}g_n(x_n) = D^{(1)}f(x_n).$$

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,. . ., xn.

Applications

 * Discrete cubic splines were originally introduced as solutions of certain minimization problems.
 * They have applications in computing nonlinear splines.
 * They are used to obtain approximate solution of a second order boundary value problem.
 * Discrete interpolatory splines have been used to construct biorthogonal wavelets.