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UBC CS TR-77-01 Summary

On the Invariance of the Interpolation Points of the Discrete l1-approximation, February 1977 Uri Ascher, 14 pages

Consider discrete $l_{1}$-approximations to a data function $f$, on some finite set of points $X$, by functions from a linear space of dimension $m < \infty$. It is known that there always exists a best approximation which interpolates $f$ on a subset of $m$ points of $X$. This does not generally hold for the ``continuous'' $L_{1}$-approximation on an interval, as we show by means of an example.

We investigate the invariance of the interpolation points of the discrete $l_{1}$-approximation under a change in the approximated function. Conditions are given, under which the interpolant to a function $g$ on a set of ``best $l_{1}$ points'' of a function $f$ is a best $l_{1}$-approximant to $g$. Additional results are then obtained for the particular case of spline $l_{1}$-approximation.

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