Abstract
A general definition of recovering set for the class of integrable functions is introduced. For every Zygmund class \(\Lambda\) on the \(p\)-adic group, the existence of such sets is proved, and procedures for the complete recovery of a function \(f \in \Lambda\) and its Fourier coefficients in the Vilenkin–Chrestenson system from the values of \(f\) on one of these sets are given.
We also study the more general case in which \(p\)-adic measures or general Vilenkin–Chrestenson series rather than \(L^1\)-functions are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. G. Plotnikov, “Recovery of integrable functions and trigonometric series,” Sb. Math. 212 (6), 843–858 (2021).
V. I. Bogachev, Measure Theory (Springer, Berlin, 2007), Vol. 1.
B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh Series and Transforms. Theory and Applications (Kluwer Academic Publishers Group, Dordrecht, 1991).
F. Schipp, W. R. Wade, and P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis (Academiai Kiado, Budapest, 1990).
V. A. Skvortsov, “Henstock–Kurzweil type integrals in \(p\)-adic harmonic analysis,” Acta Math. Acad. Paedagog. Nyházi (N. S.) 20, 207–224 (2004).
M. Plotnikov, “On the Vilenkin–Chrestenson systems and their rearrangements,” J. Math. Anal. Appl. 492 (1) (2020).
V. Shapiro, “\(U(\varepsilon)\)-sets for Walsh series,” Proc. Amer. Math. Soc. 16, 867–870 (1965).
A. V. Bakhshetsyan, “Zeros of series with respect to the Rademacher system,” Math. Notes 33 (2), 84–90 (1983).
G. G. Gevorkyan, “On sets of uniqueness for Haar and Walsh systems,” Dokl. AN Armyan. SSR 73 (2), 91–96 (1981).
M. G. Plotnikov, “Multiple Walsh series and Zygmund sets,” Math. Notes 95 (5), 686–696 (2014).
N. K. Bari, A Treatise in Trigonometric Series (The Macmillan Company, New York, 1964), Vol. I, II.
A. Zygmund, Trigonometric Series, 3rd ed. (Cambridge University Press, Cambridge, 2002), Vol. I, II.
M. Plotnikov, “\(V\)-sets in the products of zero-dimensional compact Abelian groups,” Eur. Math. J. 5, 223–240 (2019).
M. G. Plotnikov, “Analysis on \(p\)-adic groups,” in Modern Problems of the Theory of Functions and Their Applications, Materials of the 20th International Saratov Winter School (Izd. “Nauchnaya Kniga”, Saratov, 2020), pp. 311–318 [in Russian].
Funding
The work of the first author was financially supported by the Russian Foundation for Basic Research under grant 20-01-00584.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 867–878 https://doi.org/10.4213/mzm13564.
Rights and permissions
About this article
Cite this article
Plotnikov, M.G., Astashonok, V.S. Recovery of Functions on \(p\)-Adic Groups. Math Notes 112, 955–964 (2022). https://doi.org/10.1134/S0001434622110293
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622110293