# Interpolation spline

A spline

$$S _ {m} ( \Delta _ {n} ; x ) = \ a _ {0} + a _ {1} x + \dots + a _ {m-1} x ^ {m-1} + \sum _ { k= 0} ^ { n-1 } C _ {k} ( x - x _ {k} ) _ {+} ^ {m} ,$$

where

$$t _ {+} = \ \left \{ \begin{array}{ll} t & t \geq 0 , \\ 0 & t < 0 , \\ \end{array} \ \ x _ {0} < \dots < x _ {n} , \right .$$

that coincides with a given function at given distinct points $\{ \overline{x}\; _ {i} \}$. For $m = 2 k + 1$ one usually takes $\overline{x}\; _ {i} = x _ {i}$, $i = 0 \dots n$, and since for $S _ {2k+1} ( \Delta _ {n} ; x )$ there still are $2 k$ free parameters, one prescribes $k$ additional conditions at $x _ {0}$ and $x _ {n}$, e.g. $S _ {2k+1} ^ {( j)} ( \Delta _ {n} ; z ) = y _ {z} ^ {( j)}$, $j = 1 \dots k$, $z = x _ {0} , x _ {n}$, where $y _ {z} ^ {( j)}$ are given numbers. If the $y _ {z} ^ {( j)}$ linearly depend on the given function, then the corresponding spline linearly depends on this function. For $m = 2 k$ one usually takes $\overline{x}\; _ {0} = x _ {0}$, $\overline{x}\; _ {n} = x _ {n}$ and $x _ {i} = ( \overline{x}\; _ {i-1} + \overline{x}\; _ {i} ) / 2$, $i = 1 \dots n - 1$, and $k$ additional conditions are prescribed at $x _ {0}$ and $x _ {n}$. If the spline $S _ {m} ( \Delta _ {n} ; x )$ has an $( m - s )$- th continuous and an $( m - s + 1 )$- st discontinuous derivative at $x _ {1} \dots x _ {n-1}$, then for $s \geq 2$ the first $( s - 1 )$- 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 $L$- and $L _ {q}$- 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)

 [a1] C. de Boor, "Splines as linear combinations of $B$-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)