If I have three coupled equations
\begin{equation}\label{22} \rho \ \xi_{,t} = \tau_{21,1} + \tau_{23,3}, \end{equation}
\begin{equation}\label{23} \tau_{21,t} = c_{66} \ \xi_{,1} + c_{46} \ \xi_{,3}, \end{equation}
\begin{equation}\label{24} \tau_{23,t} = c_{46} \ \xi_{,1} + c_{44} \ \xi_{,3}. \end{equation}
and I want to solve them by using Fourier transforms defined as
\begin{equation} \check \xi(\omega, \underline x) = \int \xi_2( t, x_1, x_3) \ e^{i\omega t} \ dt, \end{equation} \begin{equation} \check \tau_{21}(\omega, \underline x) = \int \tau_{21}( t, x_1, x_3) \ e^{i\omega t} \ dt, \end{equation}
\begin{equation} \check \tau_{23}(\omega, \underline x) = \int \tau_{23}( t,x_1, x_3) \ e^{i\omega t} \ dt . \end{equation}
Can I change the sign convention in one of the Fourier transforms, for example, solving with
\begin{equation} \check \xi(\omega, \underline x) = \int \xi_2( t, x_1, x_3) \ e^{i\omega t} \ dt, \end{equation} \begin{equation} \check \tau_{21}(\omega, \underline x) = \int \tau_{21}( t, x_1, x_3) \ e^{-i\omega t} \ dt, \end{equation}
\begin{equation} \check \tau_{23}(\omega, \underline x) = \int \tau_{23}( t,x_1, x_3) \ e^{i\omega t} \ dt . \end{equation}
where $\tau_{21}$ has changed to a negative convention whilst the other two remain as positive convention? Is this permitted?
By changing the sign for $\omega$ in computing $\check \tau_{21}(\omega, \underline x)$, you have flipped the sign of the $\omega$ axis in the transform domain compared to your other transformations. Unless $\tau_{21}(t, ...)$ is real and even with respect to the time axis, your result for $\check \tau_{21}(\omega, \underline x)$ will be inconsistent with the other $\check \tau_{nm}(\omega, \underline x)$ functions.