  We consider the class $C(T)$ of continuous real-valued functions on the
circle. For certain classes of functions naturally characterised by the
rapidity of decrease of Fourier coefficients we investigate whether it is
possible to bring families of functions in $C(T)$ into these classes by a
change of variable. This paper was originally published in Matematicheski\v{\i}
Sbornik, 181:8 (1990), 1099--1113 (Russian). The English translation, published
in Mathematics of the USSR, Sbornik, 70:2 (1991), 541--555, is to a large
extent inconsistent with the original text. Herein the author provides a
corrected translation.
