= Wavelet coefficient thresholding
{title2=$\widehat b\mapsto\widehat b\mathbf1_{\{|\widehat b|>\tau\}}$}
Hard thresholding keeps detail coefficients whose magnitude exceeds a prescribed threshold and sets the rest to zero, while retaining the coarse approximation coefficients. <Soft thresholding> instead uses $\operatorname{sgn}(\widehat b)(|\widehat b|-\tau)_+$. The <simultaneous wavelet coefficient noise bound> motivates thresholds of order $\sigma\sqrt{\log n/n}$ in a normalized Gaussian regression model with at most order $n$ details. This is a nonlinear selection rule: coefficients can be removed individually rather than discarding an entire finest resolution.
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