This paper 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 (a suitable notion of) 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 to over 300 years ago) to trend filtering (from the last 15 years). We survey these connections, and contribute some new perspectives and new results along the way.

翻译：本文审视了一类单象形片状多角度函数,这些函数被称为离散样条,其属性类似于已知的样条功能类别,但衍生物的连续性被(一个适当的)不同差异的连续性概念所取代。 如此一来,离散样条与应用数学和统计的广泛发展相联,从差异和牛顿内插(追溯到300多年前)到趋势过滤(从过去15年开始 ) 。 我们对这些连接进行了调查,并提供了一些新的视角和新结果。