In this paper the subspace of the space of Lebesgue integrable functions which consists of the functions with \(\int_{x/2}^{3x/2}{f(t)\over x-t} dt\) integrable is studied by means of atomic decomposition. This space is an analog of the space of sequences, defined via the so-called Boas-Telyakovskij condition for integrability of trigonometric series, and, in turn, is closely related to the real Hardy space \(H\) since the integral given is closely related to the Hilbert transform. It is proved in the paper that this space is isomorphic to the subspace of odd functions in \(H.\) This result as well as some others were earlier obtained by the reviewer in the paper ``On asymptotics of Fourier transform for functions of certain classes'' [Anal. Math. 19, 151-168 (1993; Zbl 0794.42006)], by means of a different approach, which is mentioned in the reviewed paper. Each approach has many points in its favour. For example, in the paper under review the results similar to those described above are obtained for Walsh series and integrals which is possible just by means of the approach elaborated in the paper.