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Foundations of measure theory, the Hausdorff-Besicovitch measure, fractal nowhere differentiable Boltzano, Weierstrass, Cantor, Riemann, Darboux, Van-der-Waerden, Koch, Sierpinski, Besicovitch, and Weierstrass-Mandelbrot functions, and also some atomic functions possessing fractal properties are considered. For the first time, the atomic-fractal functions are constructed. The theory proposed and justified is illustrated on problems of synthesis of radiating structures by means of modeling the corresponding physical processes.

fractal; antenna array; antenna theory; antenna radiation pattern; low-sidelobe antenna; array signal processing; atomic function; R-function