53 research outputs found
Construction of frames for shift-invariant spaces
We construct a sequence {\phi_i(\cdot-j)\mid j\in{\ZZ}, i=1,...,r} which
constitutes a -frame for the weighted shift-invariant space
[V^p_{\mu}(\Phi)=\Big{\sum\limits_{i=1}^r\sum\limits_{j\in{\mathbb{Z}}}c_i(j)\phi_i(\cdot-j)
\Big| {c_i(j)}_{j\in{\mathbb{Z}}}\in\ell^p_{\mu}, i=1,...,r\Big},
p\in[1,\infty],] and generates a closed shift-invariant subspace of
. The first construction is obtained by choosing functions
, , with compactly supported Fourier transforms
, . The second construction, with compactly supported
gives the Riesz basis
Boundary values of holomorphic functions and heat kernel method in translation-invariant distribution spaces
We study boundary values of holomorphic functions in translation-invariant
distribution spaces of type . New edge of the wedge
theorems are obtained. The results are then applied to represent
as a quotient space of holomorphic functions. We
also give representations of elements of via the
heat kernel method. Our results cover as particular instances the cases of
boundary values, analytic representations, and heat kernel representations in
the context of the Schwartz spaces , , and
their weighted versions.Comment: 21 pages; with minor correction
On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution
We develop a convolution theory for quasianalytic ultradistributions of
Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and
use it as a base for the parametrix method in the study of new topological and
structural properties of several quasianalytic spaces of functions and
ultradistributions. In particular, our results apply to Fourier hyperfunctions
and Fourier ultra-hyperfunctions.Comment: 37 page
New distribution spaces associated to translation-invariant Banach spaces
We introduce and study new distribution spaces, the test function space
and its strong dual . These spaces
generalize the Schwartz spaces , ,
and their weighted versions. The construction of our new
distribution spaces is based on the analysis of a suitable
translation-invariant Banach space of distributions with continuous
translation group, which turns out to be a convolution module over a Beurling
algebra . The Banach space stands for
. We also study convolution and multiplicative
products on .Comment: 19 page
New classes of weighted H\"older-Zygmund spaces and the wavelet transform
We provide a new and elementary proof of the continuity theorem for the
wavelet and left-inverse wavelet transforms on the spaces and . We then
introduce and study a new class of weighted H\"older-Zygmund spaces, where the
weights are regularly varying functions. The analysis of these spaces is
carried out via the wavelet transform and generalized Littlewood-Paley pairs.Comment: 18 page
Local regularity in non-linear generalized functions
In this review article we present regularity properties of generalized
functions which are useful in the analysis of non-linear problems. It is shown
that Schwartz distributions embedded into our new spaces of generalized
functions, with additional properties described through the association, belong
to various classical spaces with finite or infinite type of regularities.Comment: 17 page
Suppleness of the sheaf of algebras of generalized functions on manifolds
We show that the sheaves of algebras of generalized functions and , are
open sets in a manifold , are supple, contrary to the non-suppleness of the
sheaf of distributions
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