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Research Report SRR02-006
Periodic Boxcar Deconvolution and Diophantine Approximation
Iain M. Johnstone and Marc Raimondo
Abstract:
We consider the non-parametric estimation of a periodic function
that is observed in additive Gaussian white noise after convolution
with a "boxcar", the indicator function of an interval. This is an
idealized model for the problem of recovery of noisy signals and
images observed with "motion blur". If the length of the boxcar is
rational, then certain frequencies are irretreviably lost in the
periodic model. We consider the rate of convergence of estimators
when the length of the boxcar is irrational, using classical results
on approximation of irrationals by continued fractions. A basic
question of interest is whether the minimax rate of convergence is
slower than for non-periodic problems with 1/f-like convolution
filters. The answer turns out to depend on the type and smoothness
of functions being estimated in a manner not seen with
"homogeneous" filters.
Primary AMS Classification: 62G20
Date: 15 July 2002
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