Abstract
By showing the unitarity of the Bargmann transform between the Fourier symmetric Sobolev space
$\mathcal{H}$ consisting of functions $f\in L^2(\mathbb{R})$ such that $ \| f \|^2_{\mathcal{H}} = \int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty $
and the corresponding Fock space, we find an orthonormal basis of $\mathcal{H}$. This allows us to find the reproducing kernel of $\mathcal{H}$, which is expected to be useful in e.g. the area of Fourier interpolation.
