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## ‣ Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/2008
Português

Relevância na Pesquisa

28.659348%

#Mathematical Physics#Mathematics - General Mathematics#11F85, 42C40, 47G30 (Primary)#26A33, 46F10 (Secondary)

In this paper a countable family of new compactly supported {\em non-Haar}
$p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ is constructed. We use the wavelet
bases in the following applications: in the theory of $p$-adic
pseudo-differential operators and equations. Namely, we study the connections
between wavelet analysis and spectral analysis of $p$-adic pseudo-differential
operators. A criterion for a multidimensional $p$-adic wavelet to be an
eigenfunction for a pseudo-differential operator is derived. We prove that
these wavelets are eigenfunctions of the fractional operator. In addition,
$p$-adic wavelets are used to construct solutions of linear and semi-linear
pseudo-differential equations. Since many $p$-adic models use
pseudo-differential operators (fractional operator), these results can be
intensively used in these models.

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## ‣ $p$-Adic Haar multiresolution analysis and pseudo-differential operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/05/2007
Português

Relevância na Pesquisa

38.659348%

#Mathematical Physics#Mathematics - General Mathematics#(Primary) 11F85, 42C40, 47G30#(Secondary) 26A33, 46F10

The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We
discuss a ``natural'' refinement equation whose solution (a refinable function)
is the characteristic function of the unit disc. This equation reflects the
fact that the characteristic function of the unit disc is a sum of $p$
characteristic functions of mutually disjoint discs of radius $p^{-1}$. This
refinement equation generates a MRA. The case $p=2$ is studied in detail. Our
MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real
setting, the refinable function generating our Haar MRA is 1-periodic, which
never holds for real refinable functions. This fact implies that there exist
infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$
generated by the same Haar MRA. All of these bases are described. We also
constructed multidimensional 2-adic Haar orthonormal bases for
${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A
criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a
pseudo-differential operator is derived. We proved also that these wavelets are
eigenfunctions of the Taibleson multidimensional fractional operator. These
facts create the necessary prerequisites for intensive using our bases in
applications.

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## ‣ $p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2006
Português

Relevância na Pesquisa

38.659348%

#Mathematical Physics#Mathematics - General Mathematics#Primary 11F85, 42C40, 47G30#Secondary 26A33, 46F10

In this paper we study some problems related with the theory of
multidimensional $p$-adic wavelets in connection with the theory of
multidimensional $p$-adic pseudo-differential operators (in the $p$-adic
Lizorkin space). We introduce a new class of $n$-dimensional $p$-adic compactly
supported wavelets. In one-dimensional case this class includes the Kozyrev
$p$-adic wavelets. These wavelets (and their Fourier transforms) form an
orthonormal complete basis in ${\cL}^2(\bQ_p^n)$. A criterion for a
multidimensional $p$-adic wavelet to be an eigenfunction for a
pseudo-differential operator is derived. We prove that these wavelets are
eigenfunctions of the Taibleson fractional operator. Since many $p$-adic models
use pseudo-differential operators (fractional operator), these results can be
intensively used in applications. Moreover, $p$-adic wavelets are used to
construct solutions of linear and {\it semi-linear} pseudo-differential
equations.

Link permanente para citações: