Hardy Spaces Generated by an Integrability Condition Original Research Article

S. A. Telyakovskiı (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 Telyakovskiıʹs 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 Telyakovskiıʹs condition to several systems, including Walsh series and integrals, in a uniform way.