Sigma-delta quantization and finite frames

History and Evolution of the Density Theorem for Gabor Frames

“…For example, redundant frames offer more resilience to the effects of noise or to erasures of frame elements than do bases. Frames were introduced by Duffin and Schaeffer [57] in the context of nonharmonic Fourier series, and today they have applications in a wide range of areas, including, for example, sampling theory [1], operator theory [100], nonlinear sparse approximation [84], pseudodifferential operators [94], wavelet theory [49], wireless communications [171], data transmission with erasures [83], signal processing [24], and quantum computing [58]. A frame which provides unique representations is called a Riesz basis.…”

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

“…There appears to be two lines for the canonical dual in the second order scheme. This can be attributed to the fact that, for even and odd oversampling the so called boundary terms are quite different (see [2,3]). …”

“…While there are numerous papers on Σ∆ in the engineering literature, [4,8,10,11,15,19,20] to which this list does not begin to do justice, a paper of Daubechies and DeVore [9] created great interest in the mathematical community [12][13][14]17,21]. In particular, Benedetto, Powell and Yilmaz [1,2] recently used finite frames to describe and examine first order Σ∆ schemes, followed by Bodmann and Paulsen [6]. Of greatest interest in these papers is the convergence rates of the algorithm and convergence constants associated with different types of frames.…”

A note on finite dual frame pairs

“…Frames in finite-dimensional spaces have been studied intensively during the past decade as tools for signal and data representation; see [1,2,5,10]. Due to their computational simplicity and robustness features, one direction of focus has been on the construction of tight frames in R d and C d ; see, e.g., [2,7,8].…”