7 Mask, amplitude, and smooth-background error
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Let \((X,\Sigma,\mu)\) be a measure space, with measurable masks \(\Omega,\widehat\Omega\subseteq X\) and scalar amplitudes \(a,\widehat a\in{\mathbb C}\) (real amplitudes are a special case). Suppose a scalar jump packet is
and an estimated packet is
Theorem 7.1: Packet perturbation bound
For \(1\le p<\infty\), interpreted first as an inequality of extended \(L^p\) seminorms,
If \(\mu(\Omega)\) and \(\mu(\Omega\triangle\widehat\Omega)\) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued \(L^p\) inequality. For the pointwise supremum, and consequently also the essential \(L^\infty\) seminorm,
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Proof of Theorem 7.1
Write
Apply the triangle inequality and use
For the supremum estimate, \(|\mathbf1_\Omega-\mathbf1_{\widehat\Omega}|\le1\) pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essential supremum. Nonempty null symmetric differences need not contribute to the essential norm; the stated bound remains valid in that case and need not be sharp. Measurability justifies the \(L^p\) triangle inequality and the indicator norm formula.
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If \(s,\widehat s\) are measurable smooth-background models, the same triangle inequality gives, for finite \(p\),
This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.
Conjecture 7.2: Stable edge-packet recovery
For specified piecewise-smooth classes, contrast bounds, separation conditions, and noise models, there should exist a multiscale packet estimator whose symmetric-difference error and amplitude error converge at rates controlled independently of the smooth approximation rate. A theorem of this kind would turn the conditional known-edge result into a stable data-driven reconstruction theorem. No universal version is asserted here.
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