Superoscillating Sequences and Supershifts for Families of Generalized Functions
Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa, Alain Yger · Complex Analysis and Operator Theory · 2022
Abstract We construct a large class of superoscillating sequences, more generally of $${\mathscr {F}}$$ F - supershifts , where $${\mathscr {F}}$$ F is a family of smooth functions in ( t , x ) (resp. distributions in ( t , x ), or hyperfunctions in x depending on the parameter t ) indexed by $$\lambda \in {\mathbb {R}}$$ λ ∈ R . The frame in which we introduce such families is that of the evolution through Schrödinger equation $$(i\partial /\partial t - {\mathscr {H}}(x))(\psi )=0$$ ( i ∂ / ∂ t - H ( x ) ) ( ψ ) = 0 ( $${\mathscr {H}}(x) = -(\partial ^2/\partial x^2)/2 + V(x)$$ H ( x ) = - ( ∂ 2 / ∂ x 2 ) / 2 + V ( x ) ), V being a suitable potential). If $${\mathscr {F}}= \{(t,x) \mapsto \varphi _\lambda (t,x)\,;\, \lambda \in {\mathbb {R}}\}$$ F = { ( t , x ) ↦ φ λ ( t , x ) ; λ ∈ R } , where $$\varphi _\lambda $$ φ λ is evolved from the initial datum $$x\mapsto e^{i\lambda x}$$ x ↦ e i λ x , $${\mathscr {F}}$$ F - supershifts will be of the form $$\{\sum _{j=0}^N C_j(N,a) \varphi _{1-2j/N}\}_{N\ge 1}$$ {