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Left-definite variations of the classical Fourier expansion theorem. (English) Zbl 1181.47050

For a self-adjoint operator \(A\) in a Hilbert space \(H\), which is bounded from below by a positive constant, L.L.Littlejohn and R.Wellman [see, J. Differ.Equations 181, No.2, 280–339 (2002; Zbl 1008.47029)] constructed the left-definite theory.
In the paper under review, this general theory is applied to the differential operator \(Ay=-y^{\prime\prime}+k y\), \(k>0\), in \(H=L^2(a,b)\), subject to periodic boundary conditions \(y(a) = y(b)\), \(y^\prime(a) = y^\prime(b)\). For every positive integer \(n\), the left-definite space \(H_n = {\mathcal D}(A^{n/2})\) and the corresponding inner product \((f, g)_n = (A^{n/2}f,A^{n/2}g)\) on \(H_n\) are computed, where \((\cdot,\cdot)\) is the standard inner product in \(L^2(a, b)\). As a consequence, for each positive integer \(n\), explicit characterizations of the domains \({\mathcal D}(A^{n/2})\) are obtained. Moreover, the Fourier expansion theorem in each left-definite space \(H_n\) is developed.

MSC:

47F05 General theory of partial differential operators
34B24 Sturm-Liouville theory
47B25 Linear symmetric and selfadjoint operators (unbounded)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 1008.47029
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