Lecture: An Exponential Sum for Coset: Refinements and Applications

An Exponential Sum for Coset: Refinements and Applications
Guangwu Xu
University of Wisconsin-Milwaukee, USA
2013-07-02 09:00
Conference Hall 322, Science Building, Tsinghua University
(8:30~9:00am, Tea, Coffee, and Cookie)
For a nontrivial multiplicative character over a finite field, Katz established an upper bound for the magnitude of summation of the character values over a special coset of the base field (as an additive subgroup of the field in consideration). In the first part of the talk, we shall describe a refinement of the Katz' estimation by showing that either the upper bound is achieved or the character sum is -1, for quadratic extension. The second part discusses how to use Katz' estimation to construct partial Fourier matrices that are well behaved compressed sensing matrices, in a deterministic manner. Finally, we shall use our refinement to the construction of sparse representation of signals in a union of orthonormal bases which has been a topic of some studies. This construction produces  an approximately mutually unbiased bases which is of particular interest in quantum information theory.
(This talk contains joint work with Zhiqiang Xu)