  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.
