Interpolation spline
A spline
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where
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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 ![]() |
[a2] | I.J. Schoenberg, "Cardinal spline interpolation" , SIAM (1973) |
[a3] | L.L. Schumaker, "Spline functions, basic theory" , Wiley (1981) |
Interpolation spline. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_spline&oldid=47396