Volume 27, pp. 124-139, 2007.
Left-definite variations of the classical Fourier expansion theorem
L. L. Littlejohn and A. Zettl
Abstract
In a recent paper, 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. We apply this theory and construct the sequences of left-definite Hilbert spaces $\{H_{n}\}_{n\in{\bf N}}$ and left-definite self-adjoint operators $\{A_{n}\}_{n\in{\bf N}}$ associated with the classical, regular self-adjoint boundary value problem consisting of the Fourier equation with periodic boundary conditions. As a particular consequence of our analysis, we obtain a Fourier expansion theorem in each left-definite space $H_{n}$ as well as an explicit representation of the domain of $A^{n/2}$ for each positive integer $n$.
Full Text (PDF) [238 KB], BibTeX
Key words
self-adjoint operator, Hilbert space, left-definite Hilbert space, left-definite operator, regular self-adjoint boundary value problem, Fourier series
AMS subject classifications
34B24, 33B10
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