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
where g1(x), . . ., gn(x) are polynomials of degree 3. Let h > 0. If
then g(x) 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(x) 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(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:
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.
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