Local and Global Phaseless Sampling in Real Spline Spaces

Abstract

We study the recovery of functions in real spline spaces from unsigned sampled values. We consider two types of recovery. The one is to recover functions locally from finitely many unsigned samples. And the other is to recover functions on the whole line from infinitely many unsigned samples. In both cases, we give characterizations for a sequence of distinct points to be a phaseless sampling sequence, at which any nonseparable function is determined up to a sign on an interval or on the whole line by its unsigned sampled values. Moreover, for the case of local recovery, we also study the almost phaseless sampling and give a necessary and sufficient condition for a sequence of points to admit local recovery for almost all functions.

Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

“…On the other hand, observe that for m = 1, conditions (1.1) -(1.4) coincide with [29,Theorem 1.2]. Therefore, E is a linear sampling sequence for…”

“…The paper is organized as follows. In Section 2, we generalize a result on local phaseless sampling in [29]. For the real function space generated by B-splines with arbitrary knots, we give a necessary and sufficient condition for a sequence of points to be a local phaseless sampling sequence.…”

“…In [29], we gave a characterization of phaseless sampling sequences for V m . Specifically, we gave a necessary and sufficient condition on the sampling sequence {x n : n ∈ Z} such that any nonseparable function f ∈ V m is uniquely determined up to a sign from its unsigned sampled values |f (x n )|, n ∈ Z.…”

Phase retrieval of complex and vector-valued functions

“…contains only two elements ±f , and that the whole real linear subspace (C) [3,17,19,20,39,43,52,54]. In Section 2.2, we show that a complex conjugate phase retrieval function in C has its real part being phase retrieval in (C), and hence (C) is phase retrieval if C is complex conjugate phase retrieval, see Theorem 2.5 and Corollary 2.6.…”

“…and the unitary invariant space S is phase retrieval if every vector-valued function in S is phase retrieval, see [3,9,12,17,19,20,43,52,53,54] for phase retrieval of scalar-valued functions in various function spaces. In Section 3.1, we characterize the phase retrieval of vector-valued functions in S, see Theorems 3.2 and 3.3.…”