We give a functional analysis proof that (under an additional hypothesis) a locally compact group is compact if and only if its C*-algebra is weakly compact, a theorem first established in G. A. Elliott: A characterization of compact groups, Proc. Amer. Math. Soc. 29 (1971), 621.

We introduce a discretization scheme for localized continuous frames using methods from quasi–Monte Carlo integration and discrepancy theory. We first reduce the discretization problem of the frame to a cubature problem on its phase space. By extending the classical notions of discrepancy of a point set and variation of a function to the entire euclidian space, and establishing a corresponding Koksma-Hlawka inequality, we are able to solve this cubature problem for the phase space $\mathbb{R}^d$. This approach separates localization properties of the continuous frame from the distribution properties of the sampling set, allowing for a universal usage of the sampling set.

Recently, Daubechies, DeVore, Foucart, Hanin, and Petrova introduced a univariate Riesz basis of L_2 consisting of "small" neural networks with ReLU activation. We show that this is also a Riesz basis for Sobolev and Barron classes with smoothness smaller one on the d-cube. Based on this, we easily re-prove some recent results on the approximation by NNs, without tedious local approximations and with "good" dependence on the dimension d. We also discuss how to train these NNs based on function values.

The action of classical operators such as maximal functions or the Hilbert transform on weighted Lebesgue spaces is one of the main topics of interest in harmonic analysis. In this talk we discuss a recent development of a novel, multilinear singular integral theory that incorporates matrix weights. First, we develop from scratch a theory of multilinear Muckenhoupt classes for matrix weights, using techniques inspired from convex combinatorics and differential geometry. Next, we fully characterize the action of multilinear Calderón--Zygmund operators and (sub)multilinear maximal functions on cartesian products of matrix weighted Lebesgue spaces. On the one hand, we develop new versions of standard localization techniques such as sparse domination and Reverse Hölder inequalities. On the other hand, we introduce a new concept of directional non-degeneracy for integral kernels. Thus, we generalize and unify several previous results among others due to Goldberg, Lerner–Li–Ombrosi as well as Nazarov--Petermich--Treil--Volberg, whose methods were not applicable in the multilinear, matrix weighted setting. This talk is based on joint work with Dr. Zoe Nieraeth (University of the Basque country, UPV/EHU).

It is the purpose of this note to demonstrate how time-frequency methods can be used in order to get a quantitative understanding of the rate of recovery for functions (or their Fourier transforms) from sampled versions. The approach presented reveals that (once more) the Segal algebra $S_0(R^d)$ can be used as a key player, while on the other hand certain Shubin classes $Q_s(R^d)$ allow a symmetric formulation, due to the fact that they are compactly embedded into $S_0(R^d)$ for $s > d$ and in addition also Fourier-invariant. The are useful because they form a family of Fourier invariant Banach algebras with respect to both convolution and pointwise multiplication. The key aspect is to relate the recovery from regular samples to the transition from the a-periodic phase space to a flat torus situation and to periodization in phase space. This allows to allow well-established method from the theory of Wiener Amalgam spaces over $R^d$. This manuscript offers the description of first principles, but the approach has the potential of far reaching generalizations. By addressing the problem of transition between discrete and continuous signals it also provides a contribution to the general idea of Conceptual Harmonic Analysis.