Discrete spline interpolation

Summary

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

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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.[1]

Alternative formulation 1

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

 
 
 

Alternative formulation 2

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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[1]

 

This states that the central differences   are continuous at xi.

Example

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

 

Discrete cubic spline interpolant

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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

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  • 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]

References

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  1. ^ a b c d e f Tom Lyche (1979). "Discrete Cubic Spline Interpolation". BIT. 16 (3): 281–290. doi:10.1007/bf01932270. S2CID 122300608.
  2. ^ a b Mangasarian, O. L.; Schumaker, L. L. (1971). "Discrete splines via mathematical programming". SIAM J. Control. 9 (2): 174–183. doi:10.1137/0309015.
  3. ^ 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.
  4. ^ 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  5. ^ 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)