We give a fundament for Berezin's analytic Psi do considered in [4] in terms of Bargmann images of Pilipovic spaces. We deduce basic continuity results for such Psi do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply wellknown continuity results for real Psi do with symbols in modulation spaces, when acting on other modulation spaces.

We prove that if omega(1) and omega(2) are moderate weights and B is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding i : M(omega(1), B) -> M(omega(2), B) between two modulation spaces to be compact is that the quotient omega(2)/omega(1) vanishes at infinity. Moreoverwe show, that the boundedness of omega(2)/omega(1) is a necessary and sufficient condition for the previous embedding to be continuous.

Let s ≥ 1, ω,ω ∈ P^0_{E,s} , a ∈ 􏰒\Gamma _s^(ω_0), and let B be a suitable invariant quasi-Banach function space. Then we prove that the pseudo-differential operator Op(a) is continuous from M(ω_0·ω, B) to M(ω, B).

We extend Paley-Wiener results in the Bargmann setting deduced in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.] to larger classes of power series expansions. At the same time, we deduce characterizations of all Pilipovic spaces and their distributions (and not only of low orders as in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.]).

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

We study some classes of pseudo-differential operators with symbols a admitting anisotropic exponential type growth at infinity. We deduce mapping properties for these operators on Gelfand-Shilov spaces. Moreover, we deduce algebraic and certain invariance properties of these classes.

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.

We use these results to prove that the pseudo-differential operator Op(a) is a Schatten-q operator from M^∞ to M^p and r-nuclear operator from M^∞ to M^r when a∈M^r for suitable p, q and r in (0,∞].

We deduce mixed quasi-norm estimates of Lebesgue types on semi-continuous convolutions between sequences and functions which may be periodic or possess a weaker form of periodicity in certain directions. In these directions, the Lebesgue quasi-norms are applied on the period instead of the whole axes. © Springer Nature Switzerland AG 2019.

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.

Quasi-analytic wave-front sets of distributions which correspond to the Gevrey sequence p!(s), s is an element of [1/2, 1) are defined and investigated. The propagation of singularities are deduced by considering sequences of Gaussian windowed short-time Fourier transforms of distributions which are modifications of the original distributions by suitable restriction-extension techniques. Basic micro-local properties of the new wave-fronts are thereafter established.