H\"offding's Kernels and Periodic Covariance Representations

We start with a brief survey on H\"offding's kernels, its properties, related spectral decompositions, and discuss marginal distributions of H\"offding measures. In the second part of this note, one-dimensional covariance representations are considered over compactly supported probability distributions in the class of periodic smooth functions. H\"offding's kernels are used in the construction of mixing measures whose marginals are multiples of given probability distributions, leading to optimal kernels in periodic covariance representations.