Phase retrieval in l_2(R)

Abstract

Real phase retrieval is one of the most active areas of research today. Although there are many papers on finite dimensional real phase retrieval, very little work has been done on the infinite dimensional case which is what we will address here. We will review the major results in finite dimensional real phase retrieval for vectors and
projections and then see which results extend to infinite dimensions. In particular, we will:
(1) show that Edidin’s theorem extends to infinite dimensions.
(2) show that there are vectors doing phase retrieval but their perps fail phase retrieval.
(3) show there is a family of vectors doing phase retrieval but if any vector is deleted, it fails phase retrieval.
(4) extend the notion of full spark to infinite dimensions and see that full spark families do phase retrieval.
(5) show that the families of vectors which do phase retrieval are not dense in the families of vectors in l2.
(6) show that there are families of three Riesz bases that do phase retrieval but leave open the question of where two Riesz bases can do phase retrieval.
(7) give several constructions of families of vectors that do phase retrieval in l2.