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However, it is always possible for given nonnegative spline $f = \sum a_i B_{i,k,t}$ to obtain a refined knot sequence $t'$ so that $f=:\sum_i a_i' B_{i,k,t'}$ has nonnegative B-coefficients. The only issue here is the possibility that $f$ is not strictly positive. In that case, for every maximal zero interval $[z_l\fromto z_r]$ for $f$, make sure that both endpoints appear in $t'$ at least $k-1$ times. Any other negative coefficients are bound to disappear after finitely many knot insertions nearby. %
Early work on generalized convex splines (i.e., splines whose $j$th derivative is nonnegative for some $j$) can be found in %
%Burchard74b %\refJ Burchard, H.; Extremal positive splines with applications to interpolation and approximation by generalized convex functions; \BAMS; 79; 1974; 959--963; %
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