We extend F . Holland's definition of the space of resonant classes of functions, on the real line, to the space R ( Φ p q )  ( 1 ≦ p ,  q ≦ ∞ ) of resonant classes of measures, on locally compact abelian groups. We characterize this space in terms of transformable measures and establish a realatlonship between R ( Φ p q ) and the set of positive definite functions for amalgam spaces. As a consequence we answer the conjecture posed by L. Argabright and J. Gil de Lamadrid in their work on Fourier analysis of unbounded measures.