\(c\)-frames and \(c\)-Bessel mappings (Q1954019)

Continuous frames or \(c\)-frames are generalizations of frames in a Hilbert space \(\mathbb H\). The analysis and synthesis of a signal require a frame and its dual. A frame can have more than one dual, the canonical dual being an obvious choice. The authors show that a \(c\)-Bessel mapping \(f,\) as defined in the paper, has a \(c\)-Bessel mapping \(g\) as a dual if and only if \(g\) is a sum of the canonical dual of \(f\) and a \(c\)-Bessel mapping \(l\) which weakly belongs to the null space of the preframe operator of \(f,\) i.e., \(\left<k,l\right> \) belongs to the null space of the preframe operator (synthesis operator) for all \(k\in \mathbb H\). For a separable Hilbert space with a frame indexed by a countable indexing set, this result is given in Theorem 5.6.5 [\textit{O. Christensen}, An introduction to frames and Riesz bases. Boston, MA: Birkhäuser (2003; Zbl 1017.42022)] The authors further show, by a direct computation, that the composition of the preframe operator and the analysis operator of two square norm integrable \(c\)-Bessel mappings are trace class operators.