Conjugate variables in the Fourier and Legendre transforms

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In quantum mechanics, position $\textbf{r}$ and momentum $\textbf{p}$ are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where $f^*(\textbf{x}^*)=\sup[\langle \textbf{x}, \textbf{x}^*\rangle - f(\textbf{x}) ]$, $\textbf{x}$ and $\textbf{x}^*$ are Legendre conjugates. Furthermore, $\textbf{x}^*$ is often described as the slope of the tangent line at $f(\textbf{x})$.

In physics, the conjugate relationship between $\textbf{r}$ and $\textbf{p}$ is considered equivalent to the conjugate relationship $\textbf{x}$ and $\textbf{x}^*$ such that for the Legendre transform between the Lagrangian and the Hamiltonian $\textbf{r}$ and $\textbf{p}$ are used in place of $\textbf{x}$ and $\textbf{x}^*$. To me at least, this equivalence is not at all obvious.

What is the mathematical basis for this substitution?

In other words, are both the following true for the same variables, and if so, under what conditions?
$\mathscr{F}[g(\textbf{r})]=G(\textbf{p})$
And
$f^*(\textbf{r})=\sup[\langle \textbf{r}, \textbf{p}\rangle - f(\textbf{p}) ]$

 

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Please ignore the post. I see a mistake.