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A new class of fractal functions based on specific properties of the atomic functions (AF) yr and πm (see Kravchenko, V.F. et al., J. Commun. Tech. and Electronics, vol.40, no.12, p.118-37, 1995; Zelkin, E.G. and Kravchenko, V.F., J. Commun. Tech. and Electronics, vol.46, no.8, p.829-57, 2001) is proposed and justified. New types of atomic-fractal functions (AFF) are constructed. The first part of the paper is devoted to the basics of AF and their fractal properties. The construction of AFF is also described. Then the application of AFF to solving some problems of synthesis of discrete and continuous fractal radiating structures is presented. Numerical experiments and their comparison with results obtained by D.H. Werner et. al. (see IEEE Antennas and Propagation Magazine, vol.41, no.5, p.37-59, 1999) prove the efficiency of the new class of AFF. The proposed method of construction of self-similar AFF can cause a relatively high level of sidelobes. At the same time, it is robust to errors of the distribution of elements and their failures. In practice, this allows one to combine the benefits of equally-spaced and random antenna arrays. The method proposed and justified in this work can be applied for synthesis of a wide class of both equally and non-equally spaced (two- and three-dimensional) fractal antenna arrays.