Signal Representations Via SIP p-frames and SIP Bessel Multipliers in Separable Banach Spaces

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous [Formula: see text] spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) [Formula: see text]-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where [Formula: see text]. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP [Formula: see text]-Bessel multipliers are investigated. Moreover, as special cases, characterizations of [Formula: see text]-Riesz bases related to signal representations are given, and the multipliers for [Formula: see text]-Riesz bases are discussed. We show that SIP [Formula: see text]-Bessel multipliers for [Formula: see text]-Riesz bases are invertible. Finally, the continuity of SIP [Formula: see text]-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP [Formula: see text]-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces.

Invertibility of multipliers in Hilbert C*-modules

In this paper, we present some sufficient conditions under which Bessel multipliers in Hilbert C*-modules with semi-normalized symbols are invertible and we calculate the inverses. Especially we consider the invertibility of Bessel multipliers when the elements of their symbols are positive and when their Bessel sequences are equivalent, duals, modular Riesz bases or stable under small perturbations. We show that in these cases the inverse of a Bessel multiplier can be represented as a Bessel multiplier.