= Atomic approximation on finite-dimensional spaces of continuous functions
{title2=$\left|\int f\,d\mu-\sum_it_if(w_i)\right|\leq\varepsilon\|f\|$}
A finite regular <complex measure> of total variation one can be approximated on a finite-dimensional <vector subspace> of continuous functions by a finite sum of phased point evaluations with $\sum_i|t_i|=1$. The <Krein-Milman theorem> and the <extreme points of the dual unit ball of C(K)> give weak-star approximation by convex combinations of phased point masses. A finite <norm> net of the <vector subspace> <unit ball> turns finitely many scalar approximations into one uniform estimate. Repeated nodes may be kept separate to retain the exact coefficient-magnitude sum.
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