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Backward shift invariant subspaces with applications to band preserving and phase retrieval problems
Author(s) -
Qian Tao,
Tan Lihui
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3591
Subject(s) - mathematics , linear subspace , invariant (physics) , invariant subspace , phase retrieval , laplace transform , pure mathematics , subspace topology , mathematical analysis , discrete mathematics , mathematical physics , fourier transform
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 p ( R ) , 1 ≤ p ≤ ∞ , with fg ∈ L 2 ( 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. Copyright © 2015 John Wiley & Sons, Ltd.