Geodesic
Recovery of Functions on $p$-Adic Groups
Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 867-878.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : $p$-adic group, Fourier coefficient
Keywords: Vilenkin–Chrestenson function, $p$-ary tree, quasi-measure. 
@article{MZM_2022_112_6_a6,
     author = {M. G. Plotnikov and V. S. Astashonok},
     title = {Recovery of {Functions} on $p${-Adic} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {867--878},
     publisher = {mathdoc},
     volume = {112},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a6/}
}
TY  - JOUR
AU  - M. G. Plotnikov
AU  - V. S. Astashonok
TI  - Recovery of Functions on $p$-Adic Groups
JO  - Matematičeskie zametki
PY  - 2022
SP  - 867
EP  - 878
VL  - 112
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a6/
LA  - ru
ID  - MZM_2022_112_6_a6
ER  - 
%0 Journal Article
%A M. G. Plotnikov
%A V. S. Astashonok
%T Recovery of Functions on $p$-Adic Groups
%J Matematičeskie zametki
%D 2022
%P 867-878
%V 112
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a6/
%G ru
%F MZM_2022_112_6_a6
M. G. Plotnikov; V. S. Astashonok. Recovery of Functions on $p$-Adic Groups. Matematičeskie zametki, Tome 112 (2022) no. 6, pp. 867-878. http://geodesic.mathdoc.fr/item/MZM_2022_112_6_a6/

[1] M. G. Plotnikov, “Zadachi vosstanovleniya integriruemykh funktsii i trigonometricheskikh ryadov”, Matem. sb., 212:6 (2021), 109–125 | DOI | MR | Zbl

[2] V. I. Bogachev, Osnovy teorii mery, T. 1, NITs RKhD, IKI, M.–Izhevsk, 2006 | MR

[3] B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Ryady i preobrazovaniya Uolsha: teoriya i primenenie, Nauka, M., 1987 | MR

[4] F. Schipp, W. R. Wade, P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Academiai Kiado, Budapest, 1990 | MR

[5] V. A. Skvortsov, “Henstock–Kurzweil type integrals in $P$-adic harmonic analysis”, Acta Math. Acad. Paedagog. Nyházi (N.S.), 20 (2004), 207–224 | MR | Zbl

[6] M. Plotnikov, “On the Vilenkin–Chrestenson systems and their rearrangements”, J. Math. Anal. Appl., 492:1 (2020) | DOI | MR | Zbl

[7] V. Shapiro, “$U(\varepsilon)$-sets for Walsh series”, Proc. Amer. Math. Soc., 16 (1965), 867–870 | MR | Zbl

[8] A. V. Bakhshetsyan, “O nulyakh ryadov po sisteme Rademakhera”, Matem. zametki, 33:2 (1983), 169–178 | MR | Zbl

[9] G. G. Gevorkyan, “O mnozhestvakh edinstvennosti dlya sistem Khaara i Uolsha”, Dokl. AN Armyan. SSR, 73:2 (1981), 91–96 | MR | Zbl

[10] M. G. Plotnikov, “Kratnye ryady Uolsha i mnozhestva Zigmunda”, Matem. zametki, 95:5 (2014), 750–762 | DOI | MR | Zbl

[11] H. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[12] A. Zigmund, Trigonometricheskie ryady, Mir, M., 1965 | MR | Zbl | Zbl

[13] M. Plotnikov, “$V$-sets in the products of zero-dimensional compact Abelian groups”, Eur. Math. J., 5 (2019), 223–240 | DOI | MR | Zbl

[14] M. G. Plotnikov, “Analiz na $p$-ichnykh gruppakh”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Materialy 20-oi mezhdunarodnoi Saratovskoi zimnei shkoly, Izd-vo “Nauchnaya kniga”, Saratov, 2020, 311–318