93 research outputs found
The inner kernel theorem for a certain Segal algebra
The Segal algebra is well defined for arbitrary locally
compact Abelian Hausdorff (LCA) groups . Despite the fact that it is a
Banach space it is possible to derive a kernel theorem similar to the Schwartz
kernel theorem, of course without making use of the Schwartz kernel theorem.
First we characterize the bounded linear operators from
to by distributions in . We call this the "outer kernel theorem". The "inner kernel theorem" is
concerned with the characterization of those linear operators which have
kernels in the subspace , the main subject of
this manuscript. We provide a description of such operators as regularizing
operators in our context, mapping into test functions
in , in a -to norm continuous manner. The
presentation provides a detailed functional analytic treatment of the situation
and applies to the case of general LCA groups, without recurrence to the use of
so-called Wilson bases, which have been used for the case of elementary LCA
groups. The approach is then used in order to describe natural laws of
composition which imitate the composition of linear mappings via matrix
multiplications, now in a continuous setting. We use here that in a suitable
(weak) form these operators approximate general operators. We also provide an
explanation and mathematical justification used by engineers explaining in
which sense pure frequencies "integrate" to a Dirac delta distribution
Compactness Criteria in Function Spaces
The classical criterion for compactness in Banach spaces of functions can be
reformulated into a simple tightness condition in the time-frequency domain.
This description preserves more explicitly the symmetry between time and
frequency than the classical conditions. The result is first stated and proved
for L^2(R^d), and then generalized to coorbit spaces. As special cases, we
obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces,
modulation spaces and Bargmann-Fock spaces
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