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We study a novel spline-like basis, which we name the "falling factorialbasis", bearing many similarities to the classic truncated power basis. Theadvantage of the falling factorial basis is that it enables rapid, linear-timecomputations in basis matrix multiplication and basis matrix inversion. Thefalling factorial functions are not actually splines, but are close enough tosplines that they provably retain some of the favorable properties of thelatter functions. We examine their application in two problems: trend filteringover arbitrary input points, and a higher-order variant of the two-sampleKolmogorov-Smirnov test.

The Falling Factorial Basis and Its Statistical Applications