View source for Spline interpolation

Interpolation by means of splines (cf. [[Spline|Spline]]), that is, the construction of an [[Interpolation spline|interpolation spline]] taking given values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868201.png" /> at prescribed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868202.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868203.png" />. Interpolation splines usually satisfy further conditions at the end points. E.g., for the cubic spline <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868205.png" /> is the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868206.png" />, which, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868207.png" />, consists of piecewise-cubic polynomials and has a continuous second-order derivative, one requires that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868208.png" /> and, in addition, one condition at each end point (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s0868209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682010.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682012.png" />). If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682013.png" /> are the values of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682014.png" />-periodic function, then one requires the spline to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682015.png" />-periodic also. For polynomial splines of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682016.png" />, the number of extra conditions at each end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682018.png" /> is increased by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682019.png" />. For interpolation splines of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682020.png" />, the knots of the spline (the points of discontinuity of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682021.png" />-th derivative) are usually chosen halfway between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682022.png" />, and a further <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682023.png" /> conditions are assigned at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682025.png" />.

Spline interpolation has some advantages when compared to polynomial [[Interpolation|interpolation]]. E.g., there are sequences of partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682026.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682027.png" /> and interpolation splines for which the interpolation process converges for any continuous function, provided that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682028.png" /></td> </tr></table>

Many processes of spline interpolation give the same order of approximation as the [[Best approximation|best approximation]]. Moreover, spline interpolation of some classes of differentiable functions has the property that the error does not exceed the [[Width|width]] of the corresponding class. Spline interpolation can be used to solve certain variational problems. E.g., under sufficiently general additional conditions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682030.png" />, interpolation splines satisfy the relation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682032.png" /></td> </tr></table>

This implies the existence and uniqueness of interpolation splines of odd degree, and also the simplest results on convergence:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2a)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2b)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682036.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682037.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682038.png" /> depend only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682041.png" />. For some classes of differentiable functions, the sequence of interpolation splines converges to the function to be interpolated on any sequence of partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682042.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682043.png" /> (this occurs in case (2a)–(2b)).

In addition to polynomial interpolation splines, one can also use splines of a more general form (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682044.png" />-splines or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086820/s08682045.png" />-splines). For many of these, results analogous to (1) and (2a)–(2b) also hold. For splines with defect greater than 1 one usually carries out interpolation with multiple knots.

See also [[Spline approximation|Spline approximation]].

For references see [[Spline|Spline]].



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Lyche,   L.L. Schumaker,   "On the convergence of cubic interpolating splines"  A. Meir (ed.)  A. Sharma (ed.) , ''Spline Functions and Approximation Theory'' , Birkhäuser  (1973)  pp. 169–189</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Yu.N. Subbotin,   "Interpolating splines"  Z. Cieselski (ed.)  J. Musielak (ed.) , ''Approximation Theory'' , Reidel  (1975)  pp. 221–234</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.J. Schoenberg,   "Cardinal spline interpolation" , SIAM  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.M. Prenter,   "Splines and variational methods" , Wiley  (1975)</TD></TR></table>