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On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel

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2019, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
We consider the estimation of a periodic bivariate function ƒ(⋅,⋅) based on observations from a noisy convolution model, when the kernel (blurring) function 𝘨(⋅,⋅) is unknown. However, we are able to observe 𝘨δ(⋅,⋅), a noisy version of 𝘨, at the same time, which assures the estimation. We perform the deconvolution algorithm in Fourier domain. A preliminary thresholding procedure is applied to the Fourier coefficients of observations 𝘨δ to ensure a stable inversion. We construct a hard-thresholding wavelet estimator of ƒ using band-limited wavelet bases together with compactly supported wavelet bases so that a fast estimation algorithm exists. To evaluate the performance of our estimator, we derive the lower bounds for the mean integrated squared error (𝐿2-risk) assuming that ƒ belongs to the Besov space of mixed smoothness and the kernel 𝘨 possesses certain smoothness properties. We show that the proposed wavelet estimator is adaptive and asymptotically quasi-optimal within a logarithmic factor (in the minimax sense) in a wide range of Besov balls. Furthermore, we investigate the discrete case of our deconvolution model, as it is the common case in real life. We carry out a limited simulation study and show that our estimator performs well in a finite sample setting. Finally, we extend the minimax results to the more general 𝐿𝑝-risk (1 ≤ 𝑝 < ∞), and show that our estimator is asymptotically quasi-optimal within a logarithmic factor in this case as well.
Rida Benhaddou (Advisor)
Wei Lin (Committee Member)
Chang Liu (Committee Member)
Vladimir Vinogradov (Committee Member)
101 p.

Recommended Citations

Citations

  • Liu, Q. (2019). On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802

    APA Style (7th edition)

  • Liu, Qing. On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel. 2019. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802.

    MLA Style (8th edition)

  • Liu, Qing. "On Anisotropic Functional Fourier Deconvolution Problem with Unknown Kernel." Doctoral dissertation, Ohio University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1553711028518802

    Chicago Manual of Style (17th edition)