Schatten properties, nuclearity and minimality of shift invariant spaces

We extend Feichtinger's minimality property on smallest non-trivial time-frequency shift invariant Banach spaces, to the quasi-Banach case. Analogous properties are deduced for certain matrix classes. We use these results to prove that $\operatorname{Op}(a)$ is a Schatten-$q$ operator from $M^\infty$ to $M^p$ and $r$-nuclear operator from $M^\infty$ to $M^r$ when $a\in M^r$ for suitable $p$, $q$ and $r$ in $(0,\infty ]$.Comment: Now 26 pages. Previous title: "Minimality of time-frequency shift invariant spaces and Schatten properties for quasi-Banach spaces". It now contain results on minimality of matrix classes. Versions 1 and 2 contained mistakes about Schatten results which are corrected. A section on p-nuclear operators is adde

Continuity and Schatten properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or H\"ormander classes

We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.Comment: 38 pages. Karlheinz Gr\"ochenig was one of the authors in the first version. It was then decided that he will not be one of the authors of the paper. A new Subsection 1.8 is included which informs about H\"ormander-Weyl calculus. Sections 2 and 4 are modified. arXiv admin note: text overlap with arXiv:1404.075

Paley-Wiener properties for spaces of power series expansions

We extend Paley-Wiener results in the Bargmann setting deduced earlier by the author together with E. Nabizadeh and C. Pfeuffer to larger class of power series expansions. At the same time we deduce characterisations of all Pilipovi{\'c} spaces and their distributions (and not only of low orders as in the earlier work).Comment: 19 page

Tensor products for Gelfand-Shilov and Pilipovi{\'c} distribution spaces

We show basic properties on tensor products for Gelfand-Shilov distributions and Pilipovi{\'c} distributions. This also includes the Fubbini's property of such tensor products. We also apply the Fubbini property to deduce some properties for short-time Fourier transforms of Gelfand-Shilov and Pilipovi{\'c} distributions.Comment: 18 page

Matrix parameterized pseudo-differential calculi on modulation spaces

We consider a broad matrix parameterized family of pseudo-differential calculi, containing the usual Shubin's family of pseudo-differential calculi, parameterized by real numbers. We show that continuity properties in the framework of modulation space theory, valid for the Shubin's family extend to the broader matrix parameterized family of pseudo-differential calculi.Comment: 22 pages. The paper include straight-forward extensions on already well-known results available in papers by Bayer, Cordero, Gr\"ochenig, Heil, Wahlberg and others. In versions 2 and 3: Added one reference, and extended some results to include Lebesgue exponents in the interval (0,\infty ] instead of [1,\infty]. Corrected misprint

Gabor analysis for a broad class of quasi-Banach modulation spaces

We extend the Gabor analysis in \cite{GaSa} to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norm. For such spaces we deduce invariance and embedding properties, and that the elements admit reconstructible sequence space representations using Gabor frames.Comment: 32 pages. Certain identification results on Fourier Lebesgue and modulation spaces for compactly supported elements is added in the second version. A thorough revision is done in this third version of the paper. arXiv admin note: text overlap with arXiv:1406.382

The Zak transform and Wiener estimates on Gelfand-Shilov and modulation spaces with applications to operator theory

We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these result for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by the Zak transform.Comment: 35 pages. This is the sixth version. This is thoroughly rewritten compared to earlier versions. Some Wiener estimates are pushed to separate papers. The title and content now contain some operator theory. Some more characterisations of quasi-periodic elements are presente

Images of function and distribution spaces under the Bargmann transform

We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. 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.Comment: 53 pages. This version dates from June 2016. The main different between this version with the previous January 2016 version is that Proposition 1.2 for classifications of Gelfand-Shilov distributions with STFT is extended in full range. This leads to that Section 5 could be shorten down significantl

Schatten-von Neumann properties in the Weyl calculus

Let \Op_t(a), for $t\in \mathbf R$, be the pseudo-differential operator f(x) \mapsto (2\pi)^{-n}\iint a((1-t)x+ty,\xi)f(y)e^{i\scal {x-y}\xi} dyd\xi and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in [1,\infty ]$ on $L^2$. We are especially concerned with the Weyl case (i.{}e. when $t=1/2$). We prove that if $m$ and $g$ are appropriate metrics and weight functions respectively, $h_g$ is the Planck's function, $h_g^{k/2}m\in L^p$ for some $k\ge 0$ and $a\in S(m,g)$, then \Op_t(a)\in \mathscr I_p, iff $a\in L^p$. Consequently, if $0\le \delta <\rho \le 1$ and $a\in S^r_{\rho ,\delta}$, then \Op_t(a) is bounded on $L^2$, iff $a\in L^\infty$

The pseudo-differential calculus in a Bargmann setting

We give a fundament for Berezin's analytic $\Psi$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{\'c} 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 well-known continuity results for real $\Psi$do with symbols in modulation spaces, when acting on other modulation spaces.Comment: 37 pages. In the third version we have corrected several misprints and inserted some more details. The mathematical content is the same as previous version