Matrix-valued positive definite kernels given by expansions: strict positive definiteness

Abstract

Matrix functions of the form (x,y) ∈ Ω × Ω → ∑α Aα fα (x,y) , in which Ω is a nonempty set, the Aα are positive semi-definite matrices of the same fixed order, the fα are complex-valued positive definite kernels on Ω , and the series is convergent for all x and y in Ω define matrix-valued positive definite kernels on Ω . Here, the sum may be multi-indexed, Ω may be endowed with either a topological or a metric structure, and { fα} may inherit properties attached to the setting. In this paper, we present a criterion that establishes an abstract necessary and sufficient condition in order that the kernel is strictly positive definite on Ω . We point some implications and connections of the criterion in some relevant and concrete settings in order to motivate future work on the topic. Mathematics subject classification (2020): 42A82, 42C10, 43A35.