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.
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.
- 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.
- Tom Lyche (1979). "Discrete Cubic Spline Interpolation". BIT. 16 (3): 281–290. doi:10.1007/bf01932270.
- Mangasarian, O. L.; Schumaker, L. L. (1971). "Discrete splines via mathematical programming". SIAM J. Control. 9 (2): 174–183. doi:10.1137/0309015.
- Michael A. Malcolm (April 1977). "On the computation of nonlinear spline functions". SIAM Journal on Numerical Analysis. 14 (2): 254–282. doi:10.1137/0714017.
- Fengmin Chen, Wong, P.J.Y. (Dec 2012). "Solving second order boundary value problems by discrete cubic splines". Control Automation Robotics & Vision (ICARCV), 2012 12th International Conference: 1800–1805.CS1 maint: multiple names: authors list (link)
- Averbuch, A.Z., Pevnyi, A.B., Zheludev, V.A. (Nov 2001). "Biorthogonal Butterworth wavelets derived from discrete interpolatory splines". IEEE Transactions on Signal Processing. 49 (11): 2682–2692. CiteSeerX 10.1.1.332.7428. doi:10.1109/78.960415.CS1 maint: multiple names: authors list (link)