# Difference between revisions of "Spline"

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which is defined and has continuous $(m-1)$-st | which is defined and has continuous $(m-1)$-st | ||

<!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868002.png" /> ---> | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868002.png" /> ---> | ||

− | derivative on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868003.png" /> | + | derivative on an interval $[a,b]$, |

− | + | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868003.png" /> ---> | |

− | <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/s086800/s0868008.png" /></td> </tr></table> | + | and which coincides on each interval $[x_i,x_{i+1}]$ |

− | + | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868004.png" /> ---> | |

− | where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868009.png" /> are real numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680010.png" /> is a polynomial of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680011.png" />, | + | formed by the partition $\Delta_n$: $\alpha=x_0<x_1<\cdots<x_n=b$ |

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868005.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868006.png" /> ---> | ||

+ | with a certain algebraic polynomial of degree at most $m$. | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868007.png" /> ---> | ||

+ | Splines can be represented in the following way: | ||

+ | \[ s_m(\Delta_n;x)=P_{m-1}(x) + \sum_{k=0}^{n-1}c_k (x-x_k)^m_{+},\] | ||

+ | <!-- <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/s086800/s0868008.png" /></td> </tr></table> ---> | ||

+ | where the $c_k$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s0868009.png" /> ---> | ||

+ | are real numbers, $P_{m-1}(x)$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680010.png" /> ---> | ||

+ | is a polynomial of degree at most $m-1$, | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680011.png" /> ---> | ||

+ | and $(x-t)^m_{+}=\max\left(0,(x-t)^m\right)$. | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680012.png" /> ---> | ||

+ | The points $\{x_i\}_{i=1}^{n-1}$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680013.png" /> ---> | ||

+ | are called the knots of the spline. If a spline $s_m(\Delta_n;x)$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680014.png" /> ---> | ||

+ | has a continuous $(m-k)$-th | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680015.png" /> ---> | ||

+ | derivative on $[a,b]$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680016.png" /> ---> | ||

+ | for $k\geq 1$ | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680017.png" /> ---> | ||

+ | and at the knots the $(m-k+1)$-st | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680018.png" /> ---> | ||

+ | derivative of the spline is discontinuous, then it is said to have defect $k$. | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680019.png" /> ---> | ||

+ | Besides these polynomial splines, one also considers more general splines ($L$-splines), | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680021.png" /> ---> | ||

+ | which are "tied together" from solutions of a homogeneous linear differential equation $Ly=0$, | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680022.png" /> ---> | ||

+ | splines ($L_g$-splines) | ||

+ | <!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086800/s08680024.png" /> ---> | ||

+ | with different smoothness properties at various knots, and also splines in several variables. Splines and their generalizations often occur as extremal functions when solving extremum problems, e.g. in obtaining best quadrature formulas and best numerical differentiation formulas. Splines are applied to approximate functions (see [[Spline approximation|Spline approximation]]; [[Spline interpolation|Spline interpolation]]), and in constructing approximate solutions of ordinary and partial differential equations. They can also be used to construct orthonormal systems with good convergence properties. | ||

====References==== | ====References==== |

## Revision as of 01:09, 3 May 2012

A function $s_m(\Delta_n;x)$ which is defined and has continuous $(m-1)$-st derivative on an interval $[a,b]$, and which coincides on each interval $[x_i,x_{i+1}]$ formed by the partition $\Delta_n$: $\alpha=x_0<x_1<\cdots<x_n=b$ with a certain algebraic polynomial of degree at most $m$. Splines can be represented in the following way: \[ s_m(\Delta_n;x)=P_{m-1}(x) + \sum_{k=0}^{n-1}c_k (x-x_k)^m_{+},\] where the $c_k$ are real numbers, $P_{m-1}(x)$ is a polynomial of degree at most $m-1$, and $(x-t)^m_{+}=\max\left(0,(x-t)^m\right)$. The points $\{x_i\}_{i=1}^{n-1}$ are called the knots of the spline. If a spline $s_m(\Delta_n;x)$ has a continuous $(m-k)$-th derivative on $[a,b]$ for $k\geq 1$ and at the knots the $(m-k+1)$-st derivative of the spline is discontinuous, then it is said to have defect $k$. Besides these polynomial splines, one also considers more general splines ($L$-splines), which are "tied together" from solutions of a homogeneous linear differential equation $Ly=0$, splines ($L_g$-splines) with different smoothness properties at various knots, and also splines in several variables. Splines and their generalizations often occur as extremal functions when solving extremum problems, e.g. in obtaining best quadrature formulas and best numerical differentiation formulas. Splines are applied to approximate functions (see Spline approximation; Spline interpolation), and in constructing approximate solutions of ordinary and partial differential equations. They can also be used to construct orthonormal systems with good convergence properties.

#### References

[1] | S.B. Stechkin, Yu.N. Subbotin, "Splines in numerical mathematics" , Moscow (1976) (In Russian) |

#### Comments

I.J. Schoenberg is generally acknowledged to be the "father" of splines; these functions were named and singled out for special study by him in the middle of the 1940's. Since 1960 the field of spline interpolation and approximation has grown enormously. For a reasonably complete bibliography of papers dealing with spline functions that were published before 1973, see [a4]; a valuable bibliography is also contained in [a3].

#### References

[a1] | I.J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions. Part A: On the problem of smoothing of graduation. A first class of analytic approximation formulae" Quart. Appl Math. , 4 (1946) pp. 45–99 |

[a2] | I.J. Schoenberg, "Contributions to the problem of approximation of equidistant data by analytic functions. Part B: On the problem of osculatory formulae" Quart. Appl. Math. , 4 (1946) pp. 112–141 |

[a3] | L.L. Schumaker, "Spline functions, basic theory" , Wiley (1981) |

[a4] | F. Schurer, "A bibliography on spline functions" K. Böhmer (ed.) G. Meinardus (ed.) W. Schempp (ed.) , Spline-Funktionen , B.I. Wissenschaftsverlag Mannheim (1974) pp. 315–415 |

[a5] | P.M. Prenter, "Splines and variational methods" , Wiley (1975) |

**How to Cite This Entry:**

Spline.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Spline&oldid=25865