122 research outputs found
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 is a Schatten-
operator from to and -nuclear operator from to
when for suitable , and in .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 and . 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 , 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 be the set of Schatten-von Neumann operators of order
on . We are especially concerned with the Weyl case
(i.{}e. when ). We prove that if and are appropriate metrics and
weight functions respectively, is the Planck's function, for some and , then \Op_t(a)\in \mathscr I_p, iff
. Consequently, if and , then \Op_t(a) is bounded on , iff
The pseudo-differential calculus in a Bargmann setting
We give a fundament for Berezin's analytic do considered in
\cite{Berezin71} in terms of Bargmann images of Pilipovi{\'c} spaces. We deduce
basic continuity results for such 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 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
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