On the singularities of continuous Legendre transforms

Abstract

The analytic properties of continuous Legendre transform F ( λ ) F(\lambda ) of a function f ( t ) f(t) holomorphic in an elliptical neighborhood of the real interval [ − 1 , 1 ] [ - 1,1] are investigated. It is shown to be an entire function of exponential type whose Borel transform g ( z ) g(z) has a singularity at z 0 {z_0} if and only if f ( t ) f(t) has one at t 0 {t_0} where z 0 = cosh ⁡ t 0 {z_0} = \cosh {t_0} . The proof involves a modification of "Hadamard’s argument" on multiplication of singularities. The result may also be interpreted as a statement about the second continuous Legendre transform which gives f ( t ) f(t) in terms of F ( λ ) F(\lambda ) .