As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.
Advances in Operator Theory – Springer Journals
Published: Jul 14, 2020