The shape of the error in wavelet approximation and piecewise linear interpolation

The graph of the error in wavelet approximation, when properly rescaled, is shown to converge in the Hausdorff metric to a limit set $\Gamma$. The limit set $\Gamma$ is not a graph of a function, but rather a region bounded by the graphs of multiples of a derivative of the function, depending on the first nonvanishing moment of the wavelet. A similar result is shown for piecewise linear interpolation. Higher dimensional analogs are discussed.

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Published 1 January 2000