Covariance Functions

hischapterfocusesonBochner’stheoremanditsconnectiontopositive definite functions and covariance functions. There exists a common misunderstand- ing among scientists regarding the precise relationships between these mathematical concepts. A key point is the limited scope of Bochner’s theorem, as it only applies to continuous covariance functions. It will be emphasized the practical need for a generalization of the theorem, particularly for applications involving temporal, spatial, and spatiotemporal data analysis, where correlation models often exhibit discontinuity at the origin. Ultimately, the chapter aims to clarify and expand upon the theoretical understanding of these functions.