WIAS Preprint No. 686, (2001)

On adaptive inverse estimating linear functionals of unknown smoothness in Hilbert scales


  • Goldenshluger, Alexander
  • Pereverzev, Sergei V.

2010 Mathematics Subject Classification

  • 62G05 62G20 65R30


  • adaptive estimation, Hilbert scales, inverse problems, linear functionals, minimax risk, regularization


We address the problem of estimating the value of a linear functional ⟨ ƒ,𝑥 ⟩ from random noisy observations of 𝑦 = A 𝑥 in Hilbert scales. Both the white noise and density observation models are considered. We develop an inverse estimator of ⟨ ƒ,𝑥 ⟩ which automatically adapts to unknown smoothness of 𝑥 and ƒ. It is shown that accuracy of this adaptive estimator is only by a logarithmic factor worse than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.

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