Frames have been useful in signal transmission due to the built in redundancy. In recent years, the erasure problem in data transmission has been the focus of considerable research in the case the error estimate is measured by operator (or matrix) norm. Sample results include the characterization of one-erasure optimal Parseval frames, the connection between two-erasure optimal Parseval frames and equiangular frames, and some characterization of optimal dual frames. If iterations are allowed in the reconstruction process of the signal vector, then spectral radius measurement for the error operators is more appropriate then the operator norm measurement. We obtain a complete characterization of spectrally one-uniform frames (i.e., one-erasure optimal frames with respect to the spectral radius measurement) in terms of the redundancy distribution of the frame. Our characterization relies on the connection between spectrally optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependence property, and is also closely related to the well-known concept of k-independent set. For spectrally two-uniform frames, it is necessary that the frame must be linearly connected. We conjecture that it is also necessary that a two-uniform frame must be n-independent. We confirmed this conjecture for the case when N = n+1, n+2, where N is the number of vectors in a frame for an n-dimensional Hilbert space. Additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iii iterated reconstruction to work.

Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in Rn, but very little is known about the l2(Z) case or the l2(Zd) case. We establish some basic Gabor frame theory for l2(Z) and then generalize to the l2(Zd) case.

This is a descriptive qualitative study that explored Korean and English learning for Korean transnational immigrant children living in the United States. The study design included qualitative methods. Observations of five children in a Korean language school offered information about how they were taught Korean to retain their heritage language and culture. Additional observations of two of the children in their respective local public schools offered descriptions of their experiences learning English and U.S. public school culture. Interviews with the three teachers in these classrooms, as well as with three of the children's mothers, added background information and extended the observations. A thematic analysis process led to further understanding about the differences in the three classroom learning environments and described the ways instruction was delivered, the ways the individual children demonstrated their language learning, and the cultural context in each setting. The study found that the Korean language school and English speaking elementary schools were essential for the Korean immigrant children to improve their language proficiency in two languages as well as to learn different cultural and educational expectations.