Resonance classes of measures

Abstract

We extend F. Holland's definition of the space of resonant classes of functions, on the real line, to the space R(Φpq) (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(Φpq) 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.

Keywords

• amalgam spaces
• Fourier transform of unbounded measures
• positive definite measures
• positive definite quasimeasures
• Fourier multipliers.