Stable Calculation of the Legendre Transform

Given ƒδ(x) such that ƒδ(x) − ƒ(x) < δ, an estimate [formula] of the Legendre transform Lƒ is derived such that |[formula] − Lƒ| ≤ cδ. The constant c depends on sup |D2ƒ|, where D2ƒ is an arbitrary second derivative of ƒ(x), x ∈ Rn. An example is given which shows that the critical point of ƒ(x) cannot be calculated stably from the noisy data ƒδ(x). The notion of the generalized Legendre transform is introduced and a stable method for its calculation given the noisy data is described.