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Stable calculation of the Legendre transform. (English) Zbl 0794.65091

Authors’ summary: Given \(f_ \delta(x)\) such that \(| f_ \delta(x)- f(x)| <\delta\), an estimate \(\widetilde{Lf}_ \delta\) of the Legendre transform \(Lf\) is derived such that \(| \widetilde{Lf}_ \delta- Lf|\leq c\delta\). The constant \(c\) depends on \(\sup| D^ 2 f|\), where \(D^ 2f\) is an arbitrary second derivative of \(f(x)\), \(x\in\mathbb{R}^ n\). An example is given which shows that the critical point of \(f(x)\) cannot be calculated stably from the noisy data \(f_ \delta(x)\). The notion of the generalized Legendre transform is introduced and a stable method for its calculation given the noisy data is described.
Reviewer: P.Polcar (Brno)

MSC:

65R10 Numerical methods for integral transforms
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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