Recovery of Functions on $p$-Adic Groups

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.