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.^[2]

Let x₁, x₂, . . ., x_n-1 be an increasing sequence of real numbers. Let g(x) be a piecewise polynomial defined by

${\displaystyle g(x)={\begin{cases}g_{1}(x)&x<x_{1}\\g_{i}(x)&x_{i-1}\leq x<x_{i}{\text{ for }}i=2,3,\ldots ,n-1\\g_{n}(x)&x\geq x_{n-1}\end{cases}}}$

where g₁(x), . . ., g_n(x) are polynomials of degree 3. Let h > 0. If

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

then g(x) is called a discrete cubic spline.^[1]

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

${\displaystyle g(x_{i})=f(x_{i}){\text{ for }}i=0,1,\ldots ,n.}$

${\displaystyle D^{(1)}g_{1}(x_{0})=D^{(1)}f(x_{0}).}$

${\displaystyle D^{(1)}g_{n}(x_{n})=D^{(1)}f(x_{n}).}$

This unique discrete cubic spline is the discrete spline interpolant to f(x) in the interval [x₀ - h, x_n + h]. This interpolant agrees with the values of f(x) at x₀, x₁, . . ., x_n.

• Discrete cubic splines were originally introduced as solutions of certain minimization problems.^[1][2]
• They have applications in computing nonlinear splines.^[1][3]
• They are used to obtain approximate solution of a second order boundary value problem.^[4]
• Discrete interpolatory splines have been used to construct biorthogonal wavelets.^[5]