New Bases in the Space of Square Integrable Functions on the Field of p-Adic Numbers and Their Applications

Abstract

In this paper we summarize the results obtained in some of our recent studies in the form of a series of theorems. We present new real bases of functions in L²(B_r) that are eigenfunctions of the p-adic pseudodifferential Vladimirov operator defined on a compact set B_r ⊂ ℚ_p of the field of p-adic numbers ℚ_p and on the whole ℚ_p. We demonstrate a relationship between the constructed basis of functions in L²(ℚ_p) and the basis of p-adic wavelets in L²(ℚ_p). A real orthonormal basis in the space L²(ℚ_p, u(x) d_px) of square integrable functions on ℚ_p with respect to the measure u(x) d_px is described. The functions of this basis are eigenfunctions of a pseudodifferential operator of general form with kernel depending on the p-adic norm and with measure u(x) d_px. As an application of this basis, we present a method for describing stationary Markov processes on the class of ultrametric spaces \(\mathbb{U}\) that are isomorphic and isometric to a measurable subset of the field of p-adic numbers ℚ_p of nonzero measure. This method allows one to reduce the study of such processes to the study of similar processes on ℚ_p and thus to apply conventional methods of p-adic mathematical physics in order to calculate their characteristics. As another application, we present a method for finding a general solution to the equation of p-adic random walk with the Vladimirov operator with general modified measure u(∣x∣_p) d_px and reaction source in ℤ_p.