The theory of the one-dimensional classical Hardy spaces is a very important topic of harmonic analysis and summability theory. In this chapter, we focus our investigations on the atomic decomposition of the Hardy spaces. The Hardy spaces are investigated in many books, for example in Duren , Stein [308, 309], Stein and Weiss , Lu , Uchiyama  and Grafakos . Beyond these, the Hardy spaces have been introduced for martingales as well (see e.g. Garsia , Neveu , Dellacherie and Meyer [85, 86], Long  and Weisz ).