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

Left-Definite Variations of the Classical Fourier Expansion Theorem, Part II