Extension of localisation operators to ultradistributional symbols with super-exponential growth

In the Gelfand–Shilov setting, the localisation operator A a φ 1 , φ 2 is equal to the Weyl operator whose symbol is the convolution of a with the Wigner transform of the windows φ 2 and φ 1 . We employ this fact to extend the definition of localisation operators to symbols a having very fast super-exponential growth by allowing them to be mappings from D { M p } ( R d ) into D ′ { M p } ( R d ) , where M p , p ∈ N , is a non-quasi-analytic Gevrey type sequence. By choosing the windows φ 1 and φ 2 appropriately, our main results show that one can consider symbols with growth in position space of the form exp ( exp ( l | · | q ) ) , l , q > 0 .