2021/09/03 by Lance L. Littlejohn, Littlejohn, Lance L., Edward L. Smith +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2109.01715
openalex publication_date 2021/09/03 · openalex created_date 2021/09/13 · openalex updated_date 2026/07/28
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 \Hn\n ∈ ℕ and left-definite self-adjoint operators \An\n ∈ 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 D(An/2) for all positive integers n. Furthermore, a Fourier expansion theorem is given in each left-definite space Hn.