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

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 $L^2(B_r)$ that are eigenfunctions of the $p$-adic pseudodifferential Vladimirov operator defined on a compact set $B_r\subset \mathbb Q_p$ of the field of $p$-adic numbers $\mathbb Q_p$ and on the whole $\mathbb Q_p$. We demonstrate a relationship between the constructed basis of functions in $L^2(\mathbb Q_p)$ and the basis of $p$-adic wavelets in $L^2(\mathbb Q_p)$. A real orthonormal basis in the space $L^2(\mathbb Q_p,u(x)\,d_px)$ of square integrable functions on $\mathbb Q_p$ 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 $\mathbb Q_p$ of nonzero measure. This method allows one to reduce the study of such processes to the study of similar processes on $\mathbb Q_p$ 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 $\mathbb {Z}_p$.