Riesz-Fischer maps, semi-frames and frames in rigged Hilbert spaces (Q1980936)

A frame \((e_j)_{j \in J}\) for a Hilbert space \(\mathcal{H}\) is a generalization of an orthonormal (or more generally Riesz) basis that allows for expansions \(\xi = \sum_{j \in J} c_j e_j\) for every \(\xi \in \mathcal{H}\), but where the coefficients \((c_j)_j\) need not be unique. There exist natural examples of frames where the discrete index set \(J\) is replaced with a measure space \((X,\mu)\) and sums over \(J\) are replaced with integrals over \(X\); such frames are called continuous frames, cf. [\textit{S. T. Ali} et al., Ann. Phys. 222, No. 1, 1--37 (1993; Zbl 0782.47019)] and [\textit{G. Kaiser}, A friendly guide to wavelets. Boston: Birkhäuser (1994; Zbl 0839.42011)]. The paper under review deals with a generalization of continuous frames and bases where the vectors are distribution-valued. Formally, a locally convex vector space \(\mathcal{D}\) which is continuously and densely embedded into \(\mathcal{H}\) is assumed, so that one obtains a rigged Hilbert space \[ \mathcal{D} \hookrightarrow \mathcal{H} \hookrightarrow \mathcal{D}^{\times} . \] Here \(\mathcal{D}^\times\) denotes the dual space of continuous, conjugate-linear functionals on \(\mathcal{D}\). In this context, frame and basis properties of weakly measurable maps from a measure space \((X,\mu)\) into \(\mathcal{D}^\times\) are studied. One motivation for working in this generality is that there exist genuine continuous distribution-valued bases, an example being the family of functions \(((2\pi)^{-1/2}e^{-i \lambda x})_{\lambda \in \mathbb{R}}\) in the space \(\mathcal{S}^\times(\mathbb{R})\) of tempered distributions. On the other hand, ordinary (i.e., Hilbert space-valued) continuous Riesz bases are necessarily discrete, cf. [\textit{M. S. Jakobsen} and \textit{J. Lemvig}, J. Funct. Anal. 270, No. 1, 229--263 (2016; Zbl 1326.42039)]. The present paper surveys many results from [\textit{C. Trapani} et al., J. Fourier Anal. Appl. 25, No. 4, 2109--2140 (2019; Zbl 07077736)]. Some new results are also obtained, in particular a generalization of Riesz-Fischer sequences is defined and studied in the distributional setting. For the entire collection see [Zbl 1471.47002].