View source for Interpolation spline

<!--
i0519801.png
$#A+1 = 32 n = 1
$#C+1 = 32 : ~/encyclopedia/old_files/data/I051/I.0501980 Interpolation spline
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.B. Stechkin,   Yu.N. Subbotin,   "Splines in numerical mathematics" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.H. Ahlberg,   E.N. Nilson,   J.F. Walsh,   "Theory of splines and their applications" , Acad. Press  (1967)</TD></TR></table>

====Comments====
Cf. also [[Spline|Spline]]; [[Spline approximation|Spline approximation]]; [[Spline interpolation|Spline interpolation]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. de Boor,   "Splines as linear combinations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198033.png" />-splines, a survey"  G.G. Lorentz (ed.)  C.K. Chri (ed.)  L.L. Schumaker (ed.) , ''Approximation theory'' , '''2''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.J. Schoenberg,   "Cardinal spline interpolation" , SIAM  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.L. Schumaker,   "Spline functions, basic theory" , Wiley  (1981)</TD></TR></table>