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This paper is concerned with the spectral characteristics of quaternionicpositive definite functions on the real line. We generalize the Stone's theoremto the case of a right quaternionic linear one-parameter unitary group via twodifferent types of functional calculus. From the generalized Stone's theoremswe obtain a correspondence between continuous quaternionic positive definitefunctions and spectral systems, i.e., unions of a spectral measure and aunitary anti-self-adjoint operator that commute with each other; and thendeduce that the Fourier transform of a continuous quaternionic positivedefinite function is an unusual type of quaternion-valued measure which can bedescribed equivalently in two different ways. One is related to spectralsystems (induced by the first generalized Stone's theorem), the other isrelated to non-negative finite Borel measures (induced by the secondgeneralized Stone's theorem). An application to weakly stationary quaternionicrandom processes is also presented.

Spectral characterization of quaternionic positive definite functions on the real line