Spline interpolation
Interpolation by means of splines (cf. Spline), that is, the construction of an interpolation spline taking given values at prescribed points
,
. Interpolation splines usually satisfy further conditions at the end points. E.g., for the cubic spline
, where
is the partition
, which, on
, consists of piecewise-cubic polynomials and has a continuous second-order derivative, one requires that
and, in addition, one condition at each end point (e.g.,
and
, or
and
). If the
are the values of a
-periodic function, then one requires the spline to be
-periodic also. For polynomial splines of degree
, the number of extra conditions at each end point
or
is increased by
. For interpolation splines of degree
, the knots of the spline (the points of discontinuity of the
-th derivative) are usually chosen halfway between the points
, and a further
conditions are assigned at
and
.
Spline interpolation has some advantages when compared to polynomial interpolation. E.g., there are sequences of partitions :
and interpolation splines for which the interpolation process converges for any continuous function, provided that
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Many processes of spline interpolation give the same order of approximation as the best approximation. Moreover, spline interpolation of some classes of differentiable functions has the property that the error does not exceed the width of the corresponding class. Spline interpolation can be used to solve certain variational problems. E.g., under sufficiently general additional conditions at and
, interpolation splines satisfy the relation:
![]() | (1) |
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This implies the existence and uniqueness of interpolation splines of odd degree, and also the simplest results on convergence:
![]() | (2a) |
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![]() | (2b) |
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, where the
depend only on
and
, and
. For some classes of differentiable functions, the sequence of interpolation splines converges to the function to be interpolated on any sequence of partitions
for which
(this occurs in case (2a)–(2b)).
In addition to polynomial interpolation splines, one can also use splines of a more general form (-splines or
-splines). For many of these, results analogous to (1) and (2a)–(2b) also hold. For splines with defect greater than 1 one usually carries out interpolation with multiple knots.
See also Spline approximation.
For references see Spline.
Comments
References
[a1] | T. Lyche, L.L. Schumaker, "On the convergence of cubic interpolating splines" A. Meir (ed.) A. Sharma (ed.) , Spline Functions and Approximation Theory , Birkhäuser (1973) pp. 169–189 |
[a2] | Yu.N. Subbotin, "Interpolating splines" Z. Cieselski (ed.) J. Musielak (ed.) , Approximation Theory , Reidel (1975) pp. 221–234 |
[a3] | I.J. Schoenberg, "Cardinal spline interpolation" , SIAM (1973) |
[a4] | P.M. Prenter, "Splines and variational methods" , Wiley (1975) |
Spline interpolation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spline_interpolation&oldid=11892