# Interpolation spline

A spline

where

that coincides with a given function at given distinct points . For one usually takes , , and since for there still are free parameters, one prescribes additional conditions at and , e.g. , , , where are given numbers. If the linearly depend on the given function, then the corresponding spline linearly depends on this function. For one usually takes , and , , and additional conditions are prescribed at and . If the spline has an -th continuous and an -st discontinuous derivative at , then for the first -st derivatives of the spline are prescribed at these points, requiring them to coincide with the corresponding derivatives of the function to be interpolated. Interpolation - and -splines, as well as interpolation splines in several variables, have also been considered. Interpolation splines are used to approximate a function using its values on a grid. In contrast to interpolation polynomials, there exist matrices of nodes such that the interpolation splines converge to an arbitrary given continuous integrable function.

#### References

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

[2] | J.H. Ahlberg, E.N. Nilson, J.F. Walsh, "Theory of splines and their applications" , Acad. Press (1967) |

#### Comments

Cf. also Spline; Spline approximation; Spline interpolation.

#### References

[a1] | C. de Boor, "Splines as linear combinations of -splines, a survey" G.G. Lorentz (ed.) C.K. Chri (ed.) L.L. Schumaker (ed.) , Approximation theory , 2 , Acad. Press (1976) |

[a2] | I.J. Schoenberg, "Cardinal spline interpolation" , SIAM (1973) |

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

**How to Cite This Entry:**

Interpolation spline. Yu.N. Subbotin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Interpolation_spline&oldid=17062