Cantor set and interpolation / O. D. Frolkina. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. № 6. P. 26-32
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 253-258].
In 1998, Y. Benyamini published interesting results concerning interpolation
of sequences using continuous functions \mathbb R\to\mathbb R.
In particular, he proved that there exists a continuous function \mathbb R\to \mathbb R which in some sense
"interpolates" all sequences (x_n)_{n\in\mathbb Z} \in [0,1]^{\mathbb Z} "simultaneously."
In 2005, M.R. Naulin and C. Uzcátegui unifyed and generalized
Benyamini's results. In this paper, the case of topological spaces X and Y
with an abelian group acting on X is considered.
A similar problem of "simultaneous interpolation" of all "generalized sequences" using
continuous mappings X\to Y is posed. Further generalizations of
Naulin–Uncátegui theorems, in particular, multidimensional analogues of Benyamini's results are obtained.
Key words:
\mathfrak G-space, continuous mapping, interpolation, Cantor set.