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(Br) that are eigenfunctions of the p-adic pseudodifferential Vladimirov operator defined on a compact set Br ⊂ ℚp of the field of p-adic numbers ℚp and on the whole ℚp. We demonstrate a relationship between the constructed basis of functions in L2(ℚp) and the basis of p-adic wavelets in L2(ℚp). A real orthonormal basis in the space L2(ℚp, u(x) dpx) of square integrable functions on ℚp with respect to the measure u(x) dpx 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) dpx. 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 ℚp of nonzero measure. This method allows one to reduce the study of such processes to the study of similar processes on ℚp 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) dpx and reaction source in ℤp.