INTERPOLATION CLASSES AND MATRIX MEANS

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, infinity) 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.