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

• Published in: Proceedings of the Steklov Institute of Mathematics Vol. 306; no. 1; pp. 20 - 32
• Main Authors: Bikulov, A. Kh, Zubarev, A. P.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Moscow Pleiades Publishing 01.09.2019; Springer Nature B.V
• Subjects: Eigenvectors; Markov processes; Mathematical analysis; Mathematical functions; Mathematics; Mathematics and Statistics; Random walk; Wavelet
• ISSN: 0081-5438; 1531-8605
• DOI: 10.1134/S0081543819050031

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 ⊂ 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 L 2 (ℚ p ) and the basis of p -adic wavelets in L 2 (ℚ p ). A real orthonormal basis in the space L 2 (ℚ p , u ( x ) d p x ) of square integrable functions on ℚ p with respect to the measure u ( x ) d p x 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 p x . As an application of this basis, we present a method for describing stationary Markov processes on the class of ultrametric spaces 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 p x and reaction source in ℤ p .