This monograph reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation, dating back 300 years, to the recent emergence of trend filtering. In this concise yet comprehensive monograph, the author uses his recognized expertise on...