We characterize periodic elements in Gevrey classes, Gelfand-Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If q is an element of [1, infinity), omega is a suitable weight and (epsilon(E)(0))' is the set of all E -periodic elements, then we prove that the dual of M-(omega)(infinity,q) boolean AND (epsilon(E)(0))' equals M-(1/omega)(infinity,q)' boolean AND (epsilon(E)(0))' by suitable extensions of Bessel's identity. (C) 2017 Elsevier Inc. All rights reserved.

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.

We deduce continuity and Schatten–von Neumann properties for operators with matrices satisfying mixed quasi-norm estimates with Lebesgue and Schatten parameters in (0, ∞]. We use these results to deduce continuity and Schatten–von Neumann properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces, or in appropriate H ̈ormander classes. 

We prove that any linear operator with kernel in a Pilipovi{\'c} or Gelfand-Shilov spacecan be factorized by two operators in the same class. We also give links onnumerical approximations for such compositions. We apply these composition rulesto deduce estimates of singular values and establish Schatten-von Neumann propertiesfor such operators.

We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, H, and its dual, H′. We prove that for such quasi-Banach space B, there are convenient Hilbert spaces, Hk, k=1,2ss, with normalized Hermite functions as orthonormal bases and such that B lies between H1 and H1, and the latter spaces lie between H and H′.

Weconsiderabroadfamilyoftestfunctionspacesandtheirdual(distribu- tion) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic ́. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions. 

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 closed interval of the real axis. The approximating function is any Herglotz function with a generating measure that is absolutely continuous with Hölder continuous density in an arbitrary neighborhood of the approximation domain. The norm used is induced by any of the standard L^p-norms where 1 ≤ p ≤ ∞. The problem of interest is to study the convergence properties of simple Herglotz functions where the generating measures are given by finite B-spline expansions, and where the real part of the approximating functions are obtained via the Hilbert transform. In practice, such approximations are readily obtained as the solution to a finite- dimensional convex optimization problem. A constructive convergence proof is given in the case with linear B-splines, which is valid for all L^p-norms with 1 ≤ p ≤ ∞. A number of useful analytical expressions are provided regarding general B-splines and their Hilbert transforms. A typical physical application example is given regarding the passive approximation of a linear system having metamaterial characteristics. Finally, the flexibility of the optimization approach is illustrated with an example concerning the estimation of dielectric material parameters based on given dispersion data. 

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition 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 consider the counter images H_♭(R^d) and H_{0,♭}(R^d) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipovic ́ spaces S_s(R^d ) and Σ_s(R^d) when 0 < s < 1/2 and deduce their images under the Bargmann transform.