A new class of superoscillatory functions based on a generalized polar coordinate system

Is it possible for a band-limited signal to possess oscillation that is arbitrarily higher than its highest Fourier component? Common knowledge assumed that the answer is ‘No.’ Counterintuitively, it turns out that there are band-limited functions that are able to oscillate arbitrarily faster than their fastest Fourier components. These are the superoscillatory functions. Since their discovery, superoscillations have been intriguing in the world of Fourier analysis, with a vast number of applications in quantum mechanics, optics, and radar theory, among other areas. A basic aim in the literature of superoscillations is to find new types of superoscillations that will be used for such technologies. In this paper, we introduce a geometrical-based method to construct a rich class of superoscillations using the concept of directional polar coordinates, developed in this research. We investigate their basic features and show how the proposed method allows generating superoscillations with an arbitrary number of superoscillatory regions, and with an arbitrary number of variables.