Deconvolution for the Case of Multiple Characteristic Functions of Cubes in R_n.

Abstract

Explicit error bounds are exhibited for a case of deconvolution with elementary convolutors on R_n. The convolutors studied are a set of n + 1 characteristic functions of cubes (e.g., with side length SQRT j, j = 1, 2, . . ., n + 1 which operate by convolution on L^1 SET INTERSECT L^2 (R^n). For a suitable choice of the approximate identity, a set of n + 1 functions (deconvolutors) in L^2(R^n) are exhibited which restore L^1 SET INTERSECT L^2 (R^n), up to convolution with the approximate identity, from the n + 1 convolutions. For the case of the convolutors operating on L^1 SET INTERSECT L^2 SET INTERSECT L^p (R^n), 1 <= p < INFINITY, explicit bounds for the reatoration error in the norm L^p (E), E compact, are exhibited; that is, error bounds for restoration restricted to a compact subset. The motivation for this study is the digital implementation of this deconvolution for the application to signal detectors which act by integrating over cubic regions. This motivation is discussed along with remarks on the significance of the topology for signals that are implied by the notion of restoration or deconvolution.