The diagnostic of the performance of numerical methods for physical models, like those in computational fluid mechanics and other fields of continuum mechanics, rely on the preservation of statistical moments of extensive quantities. Dynamic and adaptive meshing often use interpolations to represent fields over a new set of elements and require to be conservative and moment-preserving. Denoising algorithms should not affect moment distributions of data. And numerical deltas are described using the number of moments preserved. Therefore, all these methodologies benefit from the use of moment-preserving interpolations. In this article, we review the presentation of the piecewise polynomial basis functions that provide moment-preserving interpolations, better described as the collocation basis of compact finite differences, or Z-splines. We present different applications of these basis functions that show the improvement of numerical algorithms for fluid mechanics, discrete delta functions and denoising. We also provide theorems of the extension of the properties of the basis, previously known as the Strang and Fix theory, to the case of arbitrary knot partitions.