The usual power function error estimates do not capture the true order of uniform accuracy for thin plate spline interpolation to smooth data functions in one variable. In this paper we propose a new type of power function and we show, through numerical experiments, that the error estimate based upon it does match the expected order. As a by-product of our investigations we also discover the precise form of the Peano kernel for univariate thin plate spline interpolation. In addition we demonstrate that a theoretical estimate of the decay rate of the L2¡norm of the Peano kernel is a crucial ingredient for establishing the improved error bound. We close the paper with a brief investigation of the 2¡dimensional case
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