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Periodic boxcar deconvolution and Diophantine approximation. (English) Zbl 1056.62044
Summary: We consider the nonparametric 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 nonperiodic 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.

62G05 Nonparametric estimation
62M99 Inference from stochastic processes
11K60 Diophantine approximation in probabilistic number theory
62M40 Random fields; image analysis
62G20 Asymptotic properties of nonparametric inference
65R32 Numerical methods for inverse problems for integral equations
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