Classes of superoscillating functions

In this paper, we describe how the study of longevity of superoscillating functions has developed over the last several years. Specifically, we show how the evolution of superoscillations for the Schrödinger equation for the free particle naturally lead the authors to the construction of a larger class of superoscillating functions. This basic idea, originally presented in Aharonov et al. (J Math Pures Appl 99:165–173, 2013), was subsequently generalized when different differential and convolution operators replace the Laplacian in the Schrödinger equation, and it eventually led to larger classes of superoscillating functions. In this paper, we outline this process, and we show how to extend these ideas to the case of several variables, and we summarize some recent applications of superoscillations to problems of approximation in the Schwartz spaces of functions.