Oblique Pseudoinverses of Multipliers

Abstract

Multipliers are operators on a Hilbert space that have been implicitly used in applications throughout the years. Multipliers are defined via three sequences and are mainly used in signal processing and psychoacoustics among many others. They modify a signal in three steps: analysis, term-wise scalar multiplication, and synthesis. While the study of multipliers stemmed from applications, its theory has earned an equally important place in the literature in the last few decades. One of the main questions in the theory of multipliers is the invertibility of multipliers. It is known that multipliers are not always invertible, and it has been shown that under mild conditions, the inverse of an invertible multiplier can be expressed as another multiplier of a very specific structure. In this work, we extend such result to the setting of possibly noninvertible multipliers. We solved the challenges of determining an appropriate generalized inverse to replace the inverse, and determining a proper duality relationship of the sequences of the resulting multiplier. We showed that these are nontrivial tasks that required the intricacies of the theory of frames for subspaces.