The uniform convergence of thin plate spline
interpolation in two dimensions |
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Authors: | MJD Powell |
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Institution: | (1) Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3~9EW, England , GB |
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Abstract: | Summary.
Let be a function from to that has
square
integrable second derivatives and let be the thin plate spline
interpolant
to at the points in
. We seek
bounds on the error when is in the convex
hull of
the interpolation points or when is close to at least one of
the
interpolation points but need not be in the convex hull. We find, for example,
that, if is inside a triangle whose vertices are any three
of the
interpolation points, then is bounded above by a
multiple of ,
where is the length of the longest side of the triangle and
where the
multiplier is independent of the interpolation points. Further, if
is any
bounded set in that is not a subset of a single straight
line, then we
prove that a sequence of thin plate spline interpolants converges to
uniformly on . Specifically, we require , where
is now the least upper bound on the numbers and where ,
, is
the least Euclidean
distance from to an interpolation point. Our method of
analysis applies
integration by parts and the Cauchy--Schwarz inequality to the scalar product
between second derivatives that occurs in the variational calculation of thin
plate spline interpolation.
Received November 10, 1993 / Revised version received March 1994 |
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Keywords: | Mathematics Subject Classification (1991): 65D07 |
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