Superoscillations: Realisation of quantum weak values

Abstract: Superoscillations are band-limited functions with the peculiar characteristic that they can oscillate with a frequency arbitrarily faster than their fastest Fourier component. First anticipated in different contexts, such as optics or radar physics, superoscillations have recently garnered renewed interest after more modern studies have successfully linked their properties to a novel quantum measurement theory, the weak value scheme. Under this framework, superoscillations have quickly developed into a fruitful area of mathematical study whose applications have evolved from the theoretical to the practical world. Their mathematical understanding, though still incomplete, recognises such oscillations will only arise in regions where the function is extremely small, establishing an inherent limitation to their applicability. This paper aims to provide a detailed look into the current state of research, both theoretical and practical, on the topic of superoscillations, as well as introducing the two-state vector formalism under which the weak value scheme may be realised.