Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information
\noindent The paper under review studies the reconstruction of piecewise smooth data from its spectral information. Spectral projections yield superior resolution provided the data is smooth, while the presence of jump discontinuities is responsible for the occurrence of Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this can be achieved by the technique of mollification. In the original work by \textit{D. Gottlieb} and \textit{E. Tadmor} [Prog. Sci. Comput. 6, 357--375 (1985; Zbl 0597.65099)], it has been demonstrated how to achieve formal spectral convergence in the recovering process of piecewise smooth functions by using a two-parameter family of spectral mollifiers \(\psi_{p,\theta}\). The analysis of the present paper shows that a proper choice of the parameters, specifically an adaptive choice for the degree \(p\), hides the overall strength of the method. By incorporating the distance to the discontinuities \(\theta\), along with the optimal value of \(p\), the paper ends up with an exponentially accurate recovery procedure up to the immediate neighborhood of the jump discontinuities. In addition, a proper local normalization of the spectral mollifier \(\psi_{p,\theta}\) allows for a further reduction of the error in the neighborhood of the jump discontinuities. To summarize the present modified version of the original paper, the adaptivity technique along with the numerical experiments exposed in the article provide a high resolution yet robust general-purpose recovery procedure which enables the effective post-processing of piecewise smooth data.
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