Summary: In this paper, we show, for example, that if the function \(f(z)\) is analytic in the half-plane \(\text{Im}\,z>0\) and belongs to the Hardy space \(H_ p,\;0<p\leq1\), then it can be represented in the form \[ f(z)={s!\over\pi i}\int^ \infty_ {-\infty}{h(t)\,dt\over(t-z)^ {s+1}},\;\text{Im}\,z>0, \] where \(s=[1/p]\) and \(h(t)\) is real-valued and belongs to the class \(L_ 1+L_ \infty\). Moreover, \(h(t)\) is completely characterized by means of the maximal function that measures smoothness. This paper is closely linked to the results of A. J. Miyachi, A. B. Aleksandrov, V. G. Krotov, and the author.