"From frames to weak $A$-semi-frames"Antoine, Jean-PierreAbstract From frames to weak A-semi-frames Jean-Pierre Antoine, Univ. Cath. Louvain, Belgium The question that we examine in this talk is the following: given an arbitrary element f in a Hilbert space, how to expand it into a sequence of basic elements ? = (?k), k ? ?, with ? a countable index set. We first briefly recall the familiar, easy solutions : orthonormal basis, Riesz basis, frames, upper and lower semi-frames, reproducing pairs. Next we describe a recent gener- alization, namely, weak A-frames, i.e. frames controlled by a densely defined operator A, and we compare it with the so-called lower atomic systems. Finally we introduce a further generalization, called weak lower A-semi-frames, and we study their duality properties, including the possible notion of weak upper. A-semi-frame. Concrete examples are given. Bibliography: • J-P. Antoine and P. Balazs, Frames and semi-frames, J. of Physics A: Math. Theor. 44 (2011) 205201; Corrigendum, Ibid. 44 (2011) 479501 • M. Speckbacher and P. Balazs, Reproducing pairs and the continuous non stationary Gabor transform on LCA groups with applications to representations of the affine Weyl-Heisenberg group, J. of Physics A: Math. Theor., 48 (2015) 395201. • G. Bellomonte, Continuous frames for unbounded operators, Adv. Oper. Theory, 6 (2021) 41. • G. Bellomonte and R. Corso, Frames and weak frames for unbounded operators, Adv. Com- put. Math., 46 (2020) 38. • J-P. Antoine, G. Bellomonte and C. Trapani, Weak A-frames and weak A-semi-frames, Construct. Math. Anal., 4 (2021) 104. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Antoine_2022-02_Strobl_abstract-Antoine.pdf |
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