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)

Comments

Cf. also Spline; Spline approximation; Spline interpolation.

References

[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)