From generalized Riesz bases to semi-frames and beyond
(2024) Journal of Fourier Analysis and Applications — (2024)
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- Authors
Antoine, Jean-Pierre 
- et. al.
- Abstract
- The main purpose of this paper is to review the notion of continuous frame in a Hilbert space $\H$ and their various generalizations. We follow essentially two different but related paths. The first one consists in considering continuous families of vectors that are the image of a resolution of the identity through some operator. As for the second, we consider the sesquilinear forms in Hilbert space associated to given continuous families of vectors of $\H$. Along this path, we begin introducing and discussing continuous Riesz bases. Next we turn to semi-frames, both upper and lower. On the way we point the connection between lower semi-frames and metric operators, a familiar tool in the theory of the so-called $\PT\T$-symmetric quantum mechanics. This panorama includes more recent topics like $A$-frames (i.e. frames controlled by some densely defined operator $A$), reproducing pairs and, going beyond Hilbert spaces, bases and frames in a Rigged Hilbert space (typically a space of distributions).
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Citations
Antoine, J.-P., & et al. (2024). From generalized Riesz bases to semi-frames and beyond. Journal of Fourier Analysis and Applications. Submitted. https://hdl.handle.net/2078.5/23303 (Original work published 2024)