Systems of Convolution Equations, Deconvolution, Shannon Sampling, and the Wavelet and Gabor Transforms

Casey, Stephen D.; Walnut, David F.

doi:10.1137/1036140

Cited by 53 publications

(53 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We construct below minimax lower bounds for the L p -risk, 1 ≤ p < ∞, both for the continuous model (1) and the discrete model (2). For this purpose, we define the minimax L p -risk, 1 ≤ p < ∞,…”

Section: Resultsmentioning

confidence: 99%

“…for both the continuous model (1) and the discrete model (2). Using the same arguments, it is easy to see that, for each j ≥ j 0 ,β jk − b jk are also centered Gaussian random variables with variance…”

Section: Accepted M Manuscriptmentioning

confidence: 88%

“…On the other hand, Casey & Walnut (1994) and De Canditiis & Pensky (2004Pensky ( , 2006 considered the multichannel deconvolution model which arises in signal and image processing, e.g., in LIDAR (Light Detection and Ranging) remote sensing and reconstructions of blurred images (see, e.g., Park, Dho & Kong (1997)). Using the maxiset approach, De Canditiis & Pensky (2006) derived upper bounds for the L p -risk, 1 < p < ∞, over a wide range of Besov balls, of an adaptive term-by-term thresholding wavelet estimator for a fixed target function f (·).…”

Section: Accepted M Manuscriptmentioning

confidence: 99%

“…Pensky & Sapatinas (2009) obtained minimax lower bounds and proposed an adaptive (linear or block thresholding) wavelet estimator, for both the functional deconvolution model (1) and its discrete version (2), that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, under the L 2 -risk over a wide range of Besov balls.…”

Section: Accepted M Manuscriptmentioning

confidence: 99%

“…In this section, we briefly present inverse problems discussed in Section 1 which can be seen as applications of the functional deconvolution model (1) or its discrete version (2). The optimality (in the minimax sense), or near-optimality within a logarithmic factor, for the L 2 -risk over a wide range of Besov balls in the Examples 1, 2 and 3 below have been discussed in Pensky & Sapatinas (2009) (see their Examples 4, 1, 5, respectively); here, we use the methodology presented in Sections 3 and 4 to check that the corresponding estimators are also optimal or near optimal under the L p -risk…”

Section: Examplesmentioning

confidence: 99%

See 4 more Smart Citations

Minimax convergence rates under the -risk in the functional deconvolution model

Petsa¹,

Sapatinas²

2009

Statistics & Probability Letters

View full text Add to dashboard Cite

To cite this version:Athanasia Petsa, Theofanis Sapatinas. Minimax convergence rates under the -risk in the functional deconvolution model. Statistics and Probability Letters, Elsevier, 2009, 79 (13) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A C C E P T E D M A N U S C R I P T

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Accepted M Manuscriptmentioning

confidence: 88%