Some Problems of the Canonical Harmonic Analysis on $${\mathbb Q}_p$$

In this paper we study some problems of the canonical harmonic analysis on the field $${\mathbb Q}_p$$ of $$p$$ -adic numbers. The main elements of the canonical harmonic analysis on $${\mathbb Q}_p$$ are canonical Fourier integral transforms, canonical generalized translation operators and canonical convolution products for functions on $${\mathbb Q}_p$$ . We consider various results of the canonical harmonic analysis for functions from Lebesgue spaces $$L^\rho({\mathbb Q}_p)$$ , $$1\le\rho\le\infty$$ . Basic concepts of the canonical harmonic analysis on $${\mathbb Q}_p$$ are expand to generalized functions (or distributions), among them the canonical Fourier transforms on $${\mathbb Q}_p$$ , the generalized translation operators on $${\mathbb Q}_p$$ and others. The analogues of various results of classical harmonic analysis, including analogues of the Paley-Wiener-Schwartz theorems, are proved. We introduce a canonical convolution product on $${\mathbb Q}_p$$ for usual and generalized functions and establish some of its properties.