Abstract. We show that four successive enlargements of the Sidon--Telyakovskii's class ${\cal ST}$, introduced as new integrability and $L^1$-convergence classes, are identical. For even trigonometric series, they coincide with the wellknown even classes ${\cal F}_p$, $p>1$, introduced by Fomin in 1978. For general trigonometric series, they coincide with a Fomin-type integrability class introduced by F. Móricz in 1991. It is somewhat surprising that several `different' enlargements of ${\cal ST}$ should yield only equivalent and indeed more complicated descriptions of the Fomin's and the Fomin-type classes. We also prove that the Fomin-type classes for general series, due to F. Móricz, are subclasses of $(dv^2)'$, one of the largest known integrability and $L^1$-convergence classes, and discuss other relationships between the known integrability classes. Furthermore, we show that the Fomin-type theorems for general series can be directly deduced from the original Fomin's results for even, i.e. cosine series.

AMS Subject Classification (1991): 42A16, 42A20

Received June 5, 2000, and in final form November 6, 2001. (Registered under 2866/2009.)