Let $f(x)$ be a convex function. Define $g(y)$ via the integral:
$$\mathrm{e}^{-g(y)} = \int_{-\infty}^\infty \mathrm{d}x \, \mathrm{e}^{yx-f(x)}$$
assuming that the integral converges. The domain of $f(x)$ and $g(y)$ is the entire real line.
Is $g(y)$ the Legendre transform of $f(x)$?
Some context. I found this formula reading a paper, Eq. 13, http://stacks.iop.org/1742-5468/2011/i=03/a=P03008. There it is claimed that an integration of this form yields the Legendre transform. My question is whether this formula can be used in general as an equivalent definition of the Legendre transform (for the standard definition, see https://en.wikipedia.org/wiki/Legendre_transformation).