Note on quantization for signals with bounded (r + 1)st derivative

We prove that signals with bounded (r + 1)st derivative can be quantized using a uniform c-level quantizer with the sample quantization error bounded by Bh^r+1 (c + 1 - 2^{r + 1}). Here B is the bound on ∥s^(r+1)∥_∞ and h is the sampling interval. Next, for this quantization we study the optimal choice of c that minimizes the worst case error in reconstructing s by a piecewise-polynomial function subject to a constraint that the number (log c)/h of bits per second used to represent the signal is held fixed. This optimal level is 2^r+2 for initial values of r, and is 2^r+3 for large r. Finally, we discuss a problem of representing a signal using a finite amount of memory. We prove that this quantization is almost optimal among all representations. (Here by optimal we mean a representation that allows one to reconstruct signals with the worst case error as small as possible.) This, in particular, provides an answer to the question of optimal trade-off between sampling and quantization in signal processing.