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 different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study of time evolution for superoscillations (and supershifts) has naturally lead to their extension to the case of several variables. In this paper, however, we take a different approach and use the theory of Bernstein and Lagrange approximation of analytic functions in $${\mathbb {C}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> to obtain deeper results for the several variables case. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.