| This thesis is composed of two independent parts:; Part I is on one-bit quantization of bandlimited functions, i.e., functions on the real line with compactly supported Fourier transforms. In such a scheme, a given bandlimited function x taking values in [0, 1] is represented for each sampling density λ, by a {lcub}0, 1{rcub}-sequence qλ such that convolving this sequence with an appropriately chosen filter kernel produces an approximation of the original function x˜λ which, as λ → ∞, is required to converge to x in a given functional sense. A popular example of such a scheme is sigma-delta quantization, in which representative bitstreams are produced via a symbolic dynamics associated with a nonlinear discrete dynamical system forced by the input sample sequences. This thesis presents a new framework and improved techniques for the error analysis of sigma-delta quantization. A combination of tools from analytic number theory, harmonic analysis and dynamical systems are used to sharpen the existing error estimates.; Part II is on the functional space approach in the mathematical study of image compression. This approach, inspired by various toy models for natural images and the characterizations of linear and nonlinear approximation in wavelet bases through norm equivalences, treats images as functions in suitable Besov spaces. This thesis analyzes the validity and accuracy of this approach to a further extent, and demonstrates that while this is in general a fruitful approach, it can fail or be misleading in a variety of cases. |
