  In 2002, Littlejohn and Wellman developed a general left-definite theory for
arbitrary self-adjoint operators in a Hilbert space that are bounded below by a
positive constant. Zettl and Littlejohn, in 2005, applied this general theory
to the classical second-order Fourier operator with periodic boundary boundary
conditions. In this paper, we construct sequences of left-definite Hilbert
spaces $\{H_{n}\}_{n \in \mathbb{N}}$ and left-definite self-adjoint operators
$\{A_{n}\}_{n \in \mathbb{N}}$ associated with the Fourier operator with
semi-periodic boundary conditions. We obtain explicit formulas for the domain
of the square root of the self-adjoint operator $A$ obtained from this boundary
value problem as well as explicit representations of the domains
$\mathcal{D}(A^{n/2})$ for all positive integers $n$. Furthermore, a Fourier
expansion theorem is given in each left-definite space $H_{n}$.
