An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

Abstract

In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in \( \mathcal{L}^2 (\mathbb{Q}_p^n ) \) is constructed. We also 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. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models.