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A spline
 
A spline
  
<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/i/i051/i051980/i0519801.png" /></td> </tr></table>
+
$$
 +
S _ {m} ( \Delta _ {n} ; x )  = \
 +
a _ {0} + a _ {1} x + \dots
 +
+ a _ {m-} 1 x  ^ {m-} 1 +
 +
\sum _ { k= } 0 ^ { n- }  1
 +
C _ {k} ( x - x _ {k} ) _ {+}  ^ {m} ,
 +
$$
  
 
where
 
where
  
<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/i/i051/i051980/i0519802.png" /></td> </tr></table>
+
$$
 +
t _ {+}  = \
 +
\left \{
 +
\begin{array}{ll}
 +
t  & t \geq  0 ,  \\
 +
0  & t < 0 ,  \\
 +
\end{array}
 +
\ \
 +
x _ {0}  < \dots < x _ {n} ,
 +
\right .$$
  
that coincides with a given function at given distinct points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519803.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519804.png" /> one usually takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519806.png" />, and since for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519807.png" /> there still are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519808.png" /> free parameters, one prescribes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i0519809.png" /> additional conditions at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198011.png" />, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198015.png" /> are given numbers. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198016.png" /> linearly depend on the given function, then the corresponding spline linearly depends on this function. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198017.png" /> one usually takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198021.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198022.png" /> additional conditions are prescribed at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198024.png" />. If the spline <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198025.png" /> has an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198026.png" />-th continuous and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198027.png" />-st discontinuous derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198028.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198029.png" /> the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198030.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198031.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051980/i05198032.png" />-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.
+
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====
 
====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>
 
<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====
 
====Comments====

Revision as of 22:13, 5 June 2020


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 -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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_spline&oldid=47396
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article