Filip Tomić

In this paper, we investigate several fundamental properties of the extended Gevrey classes. Furthermore, as an application, we construct wavelets whose regularity is characterized by these classes.

We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes can be introduced by using the general framework of the weighted matrices approach to ultradifferentiable functions. We examine alternative descriptions of Gelfand–Shilov spaces related to the extended Gevrey regularity and derive their nuclearity. In addition to the Fourier transform invariance property, we present their corresponding symmetric characterizations. Finally, we consider some time–frequency representations of the introduced classes of ultradifferentiable functions.

We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes fit well within the general framework of the weighted matrices approach to ultradifferentiable functions. We examine equivalent ways of introducing Gelfand-Shilov spaces related to the extended Gevrey regularity and derive their nuclearity. In addition to the Fourier transform invariance property, we present their corresponding symmetric characterizations. Finally, we consider some time-frequency representations of the introduced classes of ultradifferentiable functions.

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert [ ]-function.

In this paper we give an overview of the results concerning extended Gevrey classes. They are studied in detail by the author and his collaborators in the series of papers. Here we present an equivalent definition of such classes, recall some of their properties and briefly discuss some applications.

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example, when initial value problems are ill-posed in Gevrey settings. In this paper, we consider a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview of extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultra distributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.

Inductive and projective type sequence spaces of sub- and super-exponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of Hörmander metrics are reviewed.