Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

Abstract

We study phaseless sampling in spline spaces generated by B-splines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phase retrieval of any nonseparable function. We also study phaseless sampling in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless sampling is possible. For any function $f$ in a complex spline space, no mater it is separable or not, we show that $|f(x){|}^2$ is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.

arXiv comment24 pages

References in corpus (8)

• Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms
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• DOLPHIn - Dictionary Learning for Phase Retrieval
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Cited by in corpus (1)

• Random phaseless sampling for causal signals in shift-invariant spaces: a zero distribution perspective