Frame multipliers are an abstract version of Toeplitz operators in frame theory and consist of a composition of a multiplication operator with the analysis and synthesis operators. Whereas the boundedness properties of frame multipliers on Banach spaces associated to a frame, so-called coorbit spaces, are well understood, their invertibility is much more difficult. We show that frame multipliers with positive symbol are Banach space isomorphisms between the corresponding coorbit spaces. The results resemble the lifting theorems in the theory of Besov spaces and modulation spaces. Indeed, the application of the abstract lifting theorem to Gabor frames yields a new lifting theorem between modulation spaces. A second application to Fock spaces yields isomorphisms between weighted Fock spaces. The proof methods are taken from the theory of localized frames and existence of inverse-closed matrix algebras. This is joint work with Peter Balazs, Austrian Academy of Sciences.

Frame multipliers are operators that depend on three given sequences (two frames for a Hilbert space and one number sequence, called the symbol or the mask of the multiplier) and whose action on a signal can be described as a composition of three steps: analysis of the signal via one of the given frames (resulting in a sequence of complex numbers), multiplication of the resulting number sequence with the given mask, and synthesis with the second given frame (leading to a modified signal). Frame multipliers are generalizations of frame operators - a frame operator can be considered as a multiplier with mask (1) and for which the analysis and the synthesis sequences are the same. While frame operators are always invertible on the considered Hilbert space, frame multipliers are not necessarily invertible. In this talks, first we briefly review some sufficient conditions for invertibility of multipliers for frames with general structure (joint work with Peter Balazs). Then we consider multipliers for Gabor frames and present some results on invertibility of such multipliers (joint work with Jerielle Malonzo). Finally, we direct the consideration toward Fréchet spaces and present some results on inversion of frame multipliers on Fréchet spaces (joint work with Stevan Pilipović). The speaker acknowledges support from the Austrian Science Fund (FWF) through Project P 35846-N “Challenges in Frame Multiplier Theory”, grant DOI 10.55776/P35846.

We investigate the applicability of frame multipliers as compressive sensing measurements. We show that, under certain conditions, subsampled frame multipliers yield measurement matrices with desirable properties. To that end, we prove a general probabilistic nullspace property for arbitrary nonempty sets, that accounts for the special measurement structure induced by subsampled frame multipliers. Conditions for uniqueness of reconstruction of signals that are sparse with respect to dictionaries or, more generally, to non-linear locally Lipschitz mappings are obtained as special cases. Furthermore, we show that a frame multiplier matrix is full superregular, i.e., that all its minors are nonzero, for almost all frame symbol vectors, provided that the underlying frames are full spark and sufficiently redundant. Since Gabor frames are full spark for almost all windows, we study Gabor multipliers in more detail and are able to derive improved constants for some scenarios. Finally, our simulation results reveal that, in many instances, subsampled frame multiplier matrices exhibit the same $\ell 1$-reconstruction performance as i.i.d. Gaussian measurement matrices.

We discuss recent progress on the so-called frame set of Hermite window functions, that is, discrete point configurations which provide a Gabor frame for a distinguished Hermite window function. Gröchenig and Lyubarskii gave a sufficient density condition for their frame sets, which leads to what we call the "safety region". For rectangular lattices and Hermite windows of order 4 and higher, we enlarge this safety region by providing new points on the boundary of this region. This talk is based on joint work with Markus Faulhuber (University of Vienna) and Ilia Zlotnikov (NTNU Trondheim).

The concept of \emph{mild distributions} provides a flexible and universal framework that unifies classical distribution theory with modern techniques from Fourier and time-frequency analysis. From an applied perspective the Banach space of mild distributions is the set of all signals, whether they are discrete or continuous, period or non-periodic. Even an irregularly sampled signal can be viewed as a mild distribution, and the usual identification of periodic and discrete signals as finite sequences is best understood in this way. From a mathematical point of view the space of mild distributions is the dual of a Banach algebra of test functions inside the Fourier algebra, meanwhile known as Feichtinger's algebra $S_0(G)$. The rich collection of properties of this functor (assigning a Banach algebra of functions for each locally compact Abelian group) implies corresponding versatility of the space of mild distributions which can be seen as the dual of $S_0(G)$. It can be also identified with the subspace of all tempered distributions having a bounded spectrogram (short time Fourier transform or STFT). Thus every mild distribution has a (Fractional) Fourier transform. Dirac combs are mild distributions and Poisson's formula shows that their Fourier transforms are Dirac combs over the orthogonal subgroup. The presentation will give a short summary and display a small selection of the many application areas (in classical Fourier analysis or modern time-frequency analysis) of this space. It will also highlight the role of weak$^*$-convergence which corresponds to uniform convergence of the STFT over compact subsets of phase space.

The short-time Fourier transform (STFT) is a fundamental tool in time-frequency analysis, particularly valued for its ability to localize information in both time and frequency. In this talk, we explore under which conditions STFT multipliers coincide with classical Fourier multipliers, with special attention to the correlation between their respective symbols. We show how STFT multipliers introduce a smoothing effect compared to their Fourier counterparts. As an application, we derive necessary conditions for the continuity of anti-Wick operators. Finally, we present related results for the discrete setting: here, STFT multipliers are known as Gabor multipliers, while Fourier multipliers correspond to linear time-invariant (LTI) filters. This talk is based on joint work with E. Bastianoni, E. Cordero, H. Feichtinger, and N. Schweighofer.

We use techniques from the theory of vector-valued distributions to present characterisations of the multiplier spaces between a number of spaces of smooth functions or distributions. In particular we use the representation of these multiplication mappings by distributional kernels. The spaces we are interested in are classical spaces of L. Schwartz' theory of distributions. This is joint work with Norbert Ortner.

When a locally convex space possesses an unconditional Schauder basis, it can be represented isomorphically as a sequence space. The advantages of this are immediate, as most topological properties have explicit characterizations in this context. Although initiated by the work of Grothendieck, the systematic establishment of sequence space representations for classical function and distribution spaces started in the 1980s with the work of Valvidiva and Vogt. Only recently, a full (and explicit) table of representations was established for the spaces appearing in Schwartz’s distribution theory. For the theory of ultradistributions, establishing analogous representations becomes exceedingly more difficult. Primarily, this has two reasons: the absence of specific orthonormal bases; the lack of corresponding functional analytic machinery. In this talk, we review recent developments that overcome both problems. In particular, we establish sequence space representations for the well-known Gelfand-Shilov spaces and their multipliers. Our techniques combine the use of tools from time-frequency analysis (Gabor frames, Wilson bases) and the theory of derived functors in functional analysis.