Subpartition of unity for B-splines ($0\le {\sum }_i{N}_{i,k}(t)\le 1$)

A finite collection of the standard partition-normalized B-splines is nonnegative and has sum at most one on the entire real line. Extend the knot sequence in both directions. The full order-one collection sums to one, and summing the Cox-de Boor recurrence preserves that sum because the two coefficients of each lower-order function add to one. The finite collection is a subset. On the basic knot interval the finite sum equals one, also following from Marsden's identity.