Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm given by: $$ |c+\hat{f}|_{1\dot{+}\hat{X}}=|c|+|\hat{f}|_{\hat{X}} $$
Now my question is: for an element $f=\alpha+\hat{h}$ in $1\dot{+}\hat{X}$, what is the necessary and sufficient condition for $f$ to be invertible? I was informed that this may have something to do with the Wiener theorem, which is probably the one dealing with the invertibility of an element of the Wiener algebra consisting of functions with absolutely convergent Fourier coefficients, but I didn't see how it works. So can someone help me with this problem? Thank you!