Beyond frames: semi-frames and reproducing pairs (Q2002188)

The authors study semi-frames and reproducing pairs. A variety of results involving reproducing pairs are given with few results involving semi-frames. A result regarding upper semi-frames and lower semi-frames is given. Duality properties of the spaces generated by weakly measurable functions (pre-Hilbert spaces) are discussed and few results regarding reproducing pairs are proved. The existence of a reproducing pair and the non-uniqueness are proved, together with examples in cases of discrete frames and continuous frames. Reproducing pairs and partial inner product spaces (PIP-spaces) are discussed and some results are proved. Some results regarding reproducing pairs in genuine PIP-spaces (with lattices of Banach spaces and Hilbert spaces) are obtained. The cases of a Hilbert scale and of \(L^p\) spaces are considered and discussed through examples. For the entire collection see [Zbl 1411.00032].