arXiv subjects

N. Lovasoa Randrianarivony

Publications and source records attributed to N. Lovasoa Randrianarivony.

4 recordsLinked to original sources

arXiv2024

Declipping and the recovery of vectors from saturated measurements

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ allows for a linear and stable reconstruction of any vector $x\in H$ from the linear measurements $(\langle x,x_j\rangle)_{j\in J}$. However, there are many situations where some information in the frame coefficients is lost. In applications where one is using sensors with a fixed dynamic range, any measurement above that range is registered as the maximum, and any measurement below that range is registered as the minimum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We initiate a frame theoretic approach to saturation recovery in a similar way to what [BCE06] did for phase retrieval. We characterize when saturation recovery is possible, show optimal frames for use with saturation recovery correspond to minimal multi-fold packings in projective space, and prove that the classical frame algorithm may be adapted to this non-linear problem to provide a reconstruction algorithm.

math.FA↗

arXiv2020

Nondentable Sets in Banach Spaces

In his study of the Radon Nikodým property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set $A$ that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set $A$ (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of $A$-valued quasimartingales with sharply divergent behavior.

math.FA↗

arXiv2014

The transfer of property $(β)$ of Rolewicz by a uniform quotient map

We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(β)$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(β)$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(β)$-moduli of the same power-type even under renorming.

math.FA↗

arXiv2010

Property $(β)$ and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of $\ell_p$, $1 < p \neq 2 < \infty$, must be isomorphic to a linear quotient of $\ell_p$. We apply the geometric property $(β)$ of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case $1<p<2$. We also give a necessary condition for a Banach space to have $c_0$ as a uniform quotient.

math.FA↗