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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 waveletbases in the following applications: in the theory of $p$-adicpseudo-differential operators and equations. Namely, we study the connectionsbetween wavelet analysis and spectral analysis of $p$-adic pseudo-differentialoperators. A criterion for a multidimensional $p$-adic wavelet to be aneigenfunction for a pseudo-differential operator is derived. We prove thatthese wavelets are eigenfunctions of the fractional operator. In addition,$p$-adic wavelets are used to construct solutions of linear and semi-linearpseudo-differential equations. Since many $p$-adic models usepseudo-differential operators (fractional operator), these results can beintensively used in these models.

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