The theory of finite frames goes back to work of Riemann-Weber and Dini in the 19th century, and of Paley-Wiener and Duffin-Schaeffer in the 20th. Because of a seminal paper by Daubechies, Grossmann, and Meyer in 1986 in the context of wavelet theory, it has re-emerged in the present as an important theoretical tool and as a model for various applications in communications, radar, and the internet.

We develop the theory of finite frames, and then use it as a basis for sigma-delta quantization in Euclidean space. As such we integrate two themes: Discrete Analysis and Non-linear transformations.

Quantization is a staple in a range of applications from the construction of sensors to the latest technology in Super Audio CDs. Because of these applications, first order sigma-delta quantization schemes are constructed; and optimal quantization searches are designed. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that Sigma-Delta quantizers outperform the standard pulse code modulation (PCM) schemes. The technology requires harmonic analysis, tiling theory, dynamical systems techniques, uniform distribution discrepancy theory, and some methods from algebraic number theory.