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Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 1, Pages 1-14

Controlling the Gibbs phenomenon in noisy deconvolution of irregular multivariable input signals

1Texas Tech University, Department of Mathematics and Statistics, Lubbock 79409, TX, USA
2Katholieke Universiteit Nijmegen, Department of Mathematics, Nijmegen 6525 ED, The Netherlands

Received 1 January 1998; Revised 1 November 1998

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An example of inverse estimation of irregular multivariable signals is provided by picture restoration. Pictures typically have sharp edges and therefore will be modeled by functions with discontinuities, and they could be blurred by motion. Mathematically, this means that we actually observe the convolution of the irregular function representing the picture with a spread function. Since these observations will contain measurement errors, statistical aspects will be pertinent. Traditional recovery is corrupted by the Gibbs phenomenon (i.e., overshooting) near the edges, just as in the case of direct approximations. In order to eliminate these undesirable effects, we introduce an integral Cesàro mean in the inversion procedure, leading to multivariable Fejér kernels. Integral metrics are not sufficiently sensitive to properly assess the quality of the resulting estimators. Therefore, the performance of the estimators is studied in the Hausdorff metric, and a speed of convergence of the Hausdorff distance between the graph of the input signal and its estimator is obtained.