Rudin’s extension theorems and exponential convexity for matrix- and function-valued positive semidefinite functions

Matrix-valued (multivariate) correlation functions are increasingly used within both the statistics and machine learning communities, but their properties have been studied to a limited extent. The motivation of this paper comes from the fact that the celebrated local stationarity construction for scalar-valued correlations has not been considered for the matrix-valued case. The main reason is a lack of theoretical support for such a construction. We explore the problem of extending a matrix-valued correlation from a d-dimensional ball with arbitrary radius into the d-dimensional Euclidean space. We also consider such a problem over product spaces involving the d-dimensional ball with arbitrary radius. We then provide a useful architecture to matrix-valued local stationarity by defining the class of p-exponentially convex matrix-valued functions, and characterize such a class as scale mixtures of the d-Schoenberg kernels against certain families of measures. We exhibit bijections from such a class into the class of positive semidefinite matrix-valued functions and we extend exponentially convex matrix-valued functions from d-dimensional balls into the d-dimensional Euclidean space. We finally provide similar results for the case of function-valued correlations defined over certain Hilbert spaces.