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

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Shelkovich, V. M.

Skopina, M.

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