Hardy Spaces Generated by an Integrability Condition

S. A. Telyakovskiı̆ (1964, Izv. Akad. Nauk. SSR. Ser. Mat. 28, 1209–1236) proved an integrability condition for cosine series. No condition superior to that has been given so far. In this paper we identify the atomic structure of the Hardy type space that can be associated with this condition. As a consequence, we conclude that Telyakovskiı̆’s condition is equivalent to certain Sidon type inequalities. Then on the basis of this equivalence we show how the atomic technique can be used to extend Telyakovskiı̆’s condition to several systems, including Walsh series and integrals, in a uniform way. © 2001 Academic Press