STABLE CALCULATION OF THE LEGENDRE TRANSFORM

Given f(hook)δ(x) such that f(hook)δ(x) - f(hook)(x) < δ, an estimate [formula] of the Legendre transform Lf(hook) is derived such that |[formula] - Lf(hook)| ≤ cδ. The constant c depends on sup |D2f(hook)|, where D2f(hook) is an arbitrary second derivative of f(hook)(x), x ∈ Rn. An example is given which shows that the critical point of f(hook)(x) cannot be calculated stably from the noisy data f(hook)δ(x). The notion of the generalized Legendre transform is introduced and a stable method for its calculation given the noisy data is described. © 1993 Academic Press, Inc.