Superoscillations and supershifts in phase space: Wigner and Husimi function interpretations

Superoscillations, namely regions where a band-limited function f(x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is 'supershifted' so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the 'phase space' (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k(loc)(x) = partial derivative(x)arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k(loc) if L is small enough. These properties are illustrated numerically with two superoscillatory functions.