Parseval's identity states that the sum of squares of coefficients of the Fourier transform of a function equals the integral of the square of the function, or

$$ \sum_{-\infty}^{\infty} |c_n|^2 = (1/2\pi)\int^\pi_{-\pi} |f(x)|^2 dx $$ where the $c_i$ are the Fourier coefficients.

The Legendre-Fenchel transform can be viewed as a generalization of the Fourier transform. For a given function $f : X \rightarrow R$ over a vector space $X$ which has dual $X^{*}$, the transform $f^* : X^* \rightarrow R$ is defined as: $$ f^*(p) = \sup_{x \in X}\ \langle x, p\rangle - f(x) $$ where further $p = f'(x)$ is denoted as $x^*$. So my question is: Is there any natural generalization of Parseval's identity to relate $f^*$ and $f$ ? To be specific, I'm trying to relate quantities like $\|x-y\|$ to $\|p - q\|$ where $p = x^*, q = y^*$

isa way to make the transform isometric. Attouch and Wets have a few joint papers on these questions. Further names are Yosida and Moreau, but I'm no expert on that. $\endgroup$