Using a `local' integral representation of a matrix connection of order $n$ corresponding to an interpolation function of the same order, for each integer $n$, we can describe an injective map from the class of matrix connections of order $n$ to the class of positive $n$-monotone functions on $(0,\infty)$ and the range of this corresponding covers the class of interpolation functions of order $2n$. In particular, the space of symmetric connections is isomorphic to the space of symmetric positive $n$-monotone functions. Moreover, we show that, for each $n$, the class of $n$-connections extremely contains that of $(n+2)$-connections.
"Interpolation classes and matrix means." Banach J. Math. Anal. 9 (3) 140 - 152, 2015. https://doi.org/10.15352/bjma/09-3-10