The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function g∈L(R),1≤p≤∞, with fg∈L(R), that makes the band of fg to be within that of f if and only if g divided by an inner function related to f, belongs to some backward shift invariant subspace in relation to f. By the construction of backward shift invariant space, the solution g is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of f. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem.

%8 2016-04-01 %D 2016 %J Mathematical Methods in the Applied Sciences %P 1591-1598 %V 39 %@ 10991476 01704214 %U http://repository.um.edu.mo/handle/10692/12351 %W UM