Numerische Mathematik

, Volume 68, Issue 1, pp 107–128

# The uniform convergence of thin plate spline interpolation in two dimensions

• M.J.D. Powell

## Summary.

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 Cauchy--Schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation.

Mathematics Subject Classification (1991): 65D07