Analyticity, superoscillations and supershifts in several variables

Abstract: Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A more encompassing concept, supershifts, extends the idea of superoscillations to functions that are not sum of exponential functions. This broader notion is linked to Bernstein and Lagrange approximation of analytic functions in $\mathbb{C}^n$. Recent advancements in the theory of superoscillations and supershifts in one variable have focused on their time evolution. This paper takes a step further by expanding the notion of supershifts to include the case of several variables. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.