Hilbert space frames for a variety of industrial applications will be constructed. First, frames will be constructed for packet-based communication systems, such as the internet. Here, errors of erasure, that is, the loss of some coefficients will be addressed. The goal is to improve the overall speed and accuracy of data transmission. Next, specific frames for wireless communication systems will be explored with the goal of improving both speed and reliability. The differences in the nature of the error between these systems and packet-based systems requires different constructions. Also, frames for speech-recognition technology will be constructed with the aim to quickly recognize, distinguish and extract separate voices found in overlapped speech. Speech recognition is heavily dependent upon noise in the phase. So the specific issues of noisy phase and the loss of phase information will be addressed. Finally, Grassmannian (i.e. equiangular frames) frames will be derived symbolically and/or numerically for quantum physics, especially quantum information theory. The goal is to develop optimally distributed frames needed for quantum state tomography, quantum cryptography and foundationalissues in quantum mechanics. For each of these projects new classes of frames will be discovered, characterized and constructed.
Mathematics necessary to improve the speed and accuracy of the internet will be produced. This will also allow messages to be delivered accurately even if much of the original information is lost in transmission. In related work, specific problems related to wireless communications systems - such as cell phones will be addressed. Here it is desired to produce cell phones which do not fade and have crisp clear signals. Also the mathematics needed to separate voices in overlapped speech will be explored. The goal is to eventually have cars, appliances etc. which respond directly and quickly to voice communications. All of these projects have important applications to homeland security as well as civilian use.
