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.^[1]

Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences.^[2]

Discrete cubic splines

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]

Alternative formulation 1

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

${\displaystyle g_{i+1}(x_{i}-h)=g_{i}(x_{i}-h)}$

${\displaystyle g_{i+1}(x_{i})=g_{i}(x_{i})}$

${\displaystyle g_{i+1}(x_{i}+h)=g_{i}(x_{i}+h)}$

Alternative formulation 2

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

${\displaystyle D^{(0)}f(x)=f(x)}$

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

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

The conditions defining a discrete cubic spline are also equivalent to^[1]

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

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

Example

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

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

Discrete cubic spline interpolant

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.

Applications

• 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]