Some Problems of the Canonical Harmonic Analysis on Qₚ

In this paper we study some problems of the canonical harmonic analysis on the field Qₚ of p -adic numbers. The main elements of the canonical harmonic analysis on Qₚ are canonical Fourier integral transforms, canonical generalized translation operators and canonical convolution products for functions on Qₚ . We consider various results of the canonical harmonic analysis for functions from Lebesgue spaces L^ρ( Qₚ) , 1≤ρ≤∞ . Basic concepts of the canonical harmonic analysis on Qₚ are expand to generalized functions (or distributions), among them the canonical Fourier transforms on Qₚ , the generalized translation operators on Qₚ 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 Qₚ for usual and generalized functions and establish some of its properties.