In this report problems of synthesis 2D directional diagram (DD) of antennas with arbitrary apertures on atomic and R-functions (V.L. Rvachov functions) are considered. A new approach for 2D DD synthesis based on generalized N-th dimensional Kravchenko-Kotel’nikov Theorems is offered. The theory of R-functions allows edge effects are appear in singularity points by synthesis and analysis of 2D antenna arrays with arbitrary forms. In this work was carried out comparison with well-known the following methods: Whittaker-Kotel’nikov-Shannon – series expansion, Fourier – series expansion, Chebyshev – series with series generalized Kravchenko-Kotel’nikov-Levitan.