Multichannel boxcar deconvolution with growing number of channels

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with a boxcar-like kernel which is of particular interest in signal processing. It is known that, when the number of channels is finite, the precision of reconstruction of the response function increases as the number of channels M grow (even when the total number of observations n for all channels M remains constant) and this requires that the parameter of the channels form a Badly Approximable M -tuple. Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels M = Mn grow with the total number of observations n. However, in real-life situations, the number of channels M = Mn usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as n → ∞. Unfortunately, existing theoretical results cannot be blindly applied to accommodate the case when M = Mn → ∞ as n → ∞. This is due to the fact that, to the best of our knowledge, so far no one have studied the construction of a Badly Approximable M -tuple of a growing length on a specified interval, of a non-asymptotic length, of the real line, as M is growing. Therefore, this generalization requires non-trivial results in number theory. When M = Mn grows slowly as n increases, we develop a procedure for the construction of a Badly Approximable M -tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this M -tuple, which explicitly shows its dependence on M as M is growing. This result is further used for the evaluation of the L-risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels M which minimizes the L-risk. An earlier version of this paper was circulated as “Diophantine approximation and the problem of estimation of the initial speed of a wave on a finite interval in a stochastic setting, Technical Report TR-07-2009, Department of Mathematics and Statistics, University of Cyprus, Cyprus”. Supported in part by National Science Foundation (NSF), grant DMS-0652524. 53 54 M. Pensky and T. Sapatinas AMS 2000 subject classifications: Primary 62G05, 11K60; secondary 62G08, 35L05.