Discrete spline interpolation: Difference between revisions

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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 difference]]s are [[Continuous function|continuous]] at the knots whereas a [[Spline (mathematics)|spline]] is a piecewise polynomial such that its [[derivative]]s 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.<ref name=Tom>{{cite journal|last1=Tom Lyche|title=Discrete Cubic Spline Interpolation|journal=BIT|date=1979|volume=16|issue=3|pages=281–290|doi=10.1007/bf01932270|s2cid=122300608 }}</ref>

Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences.<ref name=Mangasarin>{{cite journal|author1=Mangasarian, O. L. |author2=Schumaker, L. L.|title=Discrete splines via mathematical programming|journal=SIAM J. Control.|date=1971|volume=9|issue=2|pages=174–183|doi=10.1137/0309015}}</ref>

==Discrete cubic splines==

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

:<math>

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:<math>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.</math>

This states that the central differences <math>D^{(j)}g(x)</math> are continuous at ''x''_''i''.

===Example===

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==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:

:<math>g(x_i) = f(x_i) \text{ for } i=0,1,\ldots, n.</math>

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:<math>D^{(1)}g_n(x_n) = D^{(1)}f(x_n).</math>

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.<ref name=Tom/><ref name= Mangasarin/>

* They have applications in computing nonlinear splines.<ref name=Tom/><ref>{{cite journal|last1=Michael A. Malcolm|title=On the computation of nonlinear spline functions|journal=SIAM Journal ofon Numerical Analysis|date=April 1977|volume=14|issue=2|pages=254-282254–282|doi=10.1137/0714017|bibcode=1977SJNA...14..254M }}</ref>

* They are used to obtain approximate solution of a second order boundary value problem.<ref>{{cite journal|last1=Fengmin Chen, Wong, P.J.Y.|title=Solving second order boundary value problems by discrete cubic splines|journal=Control Automation Robotics & Vision (ICARCV), 2012 12th International Conference|date=Dec 2012|pages=1800–1805}}</ref>

* Discrete interpolatory splines have been used to construct biorthogonal wavelets.<ref>{{cite journal|last1=Averbuch, A.Z., Pevnyi, A.B., Zheludev, V.A.|title=Biorthogonal Butterworth wavelets derived from discrete interpolatory splines|journal=Signal Processing, IEEE Transactions on Signal Processing|date=Nov 2001|volume=49|issue=11|pages=2682–2692|doi=10.1109/78.960415|bibcode=2001ITSP...49.2682A |citeseerx=10.1.1.332.7428}}</ref>

==References==