The uniform convergence of thin plate spline interpolation in two dimensions
 M.J.D. Powell
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Let \(f\) be a function from \({\cal R}^2\) to \({\cal R}\) that has square integrable second derivatives and let \(s\) be the thin plate spline interpolant to \(f\) at the points \(\{ \underline v_i : i \!=\! 1,2,\ldots,n \}\) in \({\cal R}^2\) . We seek bounds on the error \( f(\underline x)s(\underline x) \) when \(\underline x\) is in the convex hull of the interpolation points or when \(\underline x\) is close to at least one of the interpolation points but need not be in the convex hull. We find, for example, that, if \(\underline x\) is inside a triangle whose vertices are any three of the interpolation points, then \( f(\underline x)s(\underline x) \) is bounded above by a multiple of \(h\) , where \(h\) is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points. Further, if \({\cal D}\) is any bounded set in \({\cal R}^2\) that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to \(f\) uniformly on \({\cal D}\) . Specifically, we require \(h \!\rightarrow\! 0\) , where \(h\) is now the least upper bound on the numbers \(\{ d( \underline x, {\cal V} ) : \underline x \!\in\! {\cal D} \}\) and where \(d( \underline x, {\cal V} )\) , \(\underline x \!\in\! {\cal R}^2\) , is the least Euclidean distance from \(\underline x\) to an interpolation point. Our method of analysis applies integration by parts and the CauchySchwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation.
 Title
 The uniform convergence of thin plate spline interpolation in two dimensions
 Journal

Numerische Mathematik
Volume 68, Issue 1 , pp 107128
 Cover Date
 19940601
 DOI
 10.1007/s002110050051
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Keywords

 Mathematics Subject Classification (1991): 65D07
 Industry Sectors
 Authors

 M.J.D. Powell ^{(A1)}
 Author Affiliations

 A1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3~9EW, England, GB