Bicomplex frames (2001.07004v1)

Abstract: The main purpose is to introduce the so-called bicomplex (bc)-frames which is a special extension to bicomplex infinite Hilbert spaces of the classical frames. The crucial result is the characterization of bc-frames in terms of their idempotent components, giving rise to generalization of certain results to bc-frames. Although the extension is natural, many basic properties satisfied by classical frames do not remain valid for bc-frames, unless we restrict ourself to complex-valued Hilbert space on bicomplex numbers. By benefiting from insight provided by the classical frame theory, we discuss the construction of bc-frame operator and Weyl--Heisenberg bc-frames and we provide some new ones which are appropriate for bc-frames.