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

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 $H_{n}$.