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 ''x''₁, ''x''₂, . . ., ''x''_''n''-1 be an increasing sequence of real numbers. Let ''g''(''x'') be a piecewise polynomial defined by

:
g(x)=
\begin
g_1(x) & x

where ''g''₁(''x''), . . ., ''g''_''n''(''x'') are polynomials of degree 3. Let ''h'' > 0. If

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

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

Alternative formulation 1

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

:$g_(x_i-h) = g_i(x_i-h)$

:$g_(x_i) = g_i(x_i)$

:$g_(x_i+h) = g_i(x_i+h)$

Alternative formulation 2

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

:$D^f(x) = f(x)$

:$D^f(x)=\frac$

:$D^f(x)=\frac$

The conditions defining a discrete cubic spline are also equivalent to

:$D^g_(x_i)=D^g_i(x_i) \text j=0,1,2 \text i=1,2, \ldots, n-1.$

This states that the central differences $D^g(x)$ are continuous at ''x''_''i''.

Example

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

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

Discrete cubic spline interpolant

Let ''x''₀ < ''x''₁ and ''x''_''n'' > ''x''_''n''-1 and ''f''(''x'') be a function defined in the closed interval 'x''₀ - h, ''x''_''n'' + h Then there is a unique cubic discrete spline ''g''(''x'') satisfying the following conditions:

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

This unique discrete cubic spline is the discrete spline interpolant to ''f''(''x'') in the interval 'x''₀ - h, ''x''_''n'' + h This interpolant agrees with the values of ''f''(''x'') at ''x''₀, ''x''₁, . . ., ''x''_n.

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.

References

{{reflist

Interpolation
Splines (mathematics)