Analyticity and supershift with regular sampling
Fabrizio Colombo, Irene Sabadini, Daniele Carlo Struppa, Alain Yger · Complex Analysis and its Synergies · 2025
Abstract The notion of supershift (in itself a generalization of the notion of superoscillation arising in quantum mechanics) expresses the fact that the sampling of a function in an interval allows to compute the values of the function far from the interval. In this paper, we study the relation between supershift and real analyticity. We use a classical result due to Serge Bernstein to show that real analyticity for a complex-valued function implies a strong form of supershift. On the other hand, we use a parametric version of a result by Leonid Kantorovitch to show that the converse is not true. We additionally study the stability of functions that satisfy suitable supershift requirements under multiplication by complex numbers or primitivization.