Left-definite theory with applications to orthogonal polynomials

A combinatorial interpretation of the Legendre-Stirling numbers

“…However, as the authors show in [9] (see also [3], [6], [7], and [8]), it is possible to compute these spaces and inner products for several well-known self-adjoint operators for each positive integer r.…”

Boundary conditions associated with the general left-definite theory for differential operators

“…This theory has been applied to many types of self-adjoint differential operators, including those stemming from the second-order differential equations of Hermite, Legendre, Jacobi, Laguerre, and Fourier. Excellent surveys of these results are [6] and [19].…”

Hilbert scales and Sobolev spaces defined by associated Legendre functions

“…The study of the spaces generated by Legendre polynomials, which corresponds to m = 0 in the present work, has already been undertaken in [10] (see also [3] for a previous study and [5] for a more general theory covering some classical families of orthogonal polynomials). The analysis there is based on rewriting the inner product defined by the powers of the Legendre differential operator.…”

“…Note that is relevant that we demand u ∈ Y as part of the conditions for a function to be in the domain of the operator but once that is done, we only require the image of the differential operator to be in L 2 (−1, 1). In this sense, the cases m ≥ 1 differ in an essential way from the case m = 0, where some additional boundary conditions appear in the domain of the differential operator that are not in the 'energy space' (see Appendix B and [5] and references therein). Consider the space…”

“…We will clarify this point in the following section. We first remark that the study of the Hilbert scales defined by Legendre polynomials (and actually by some other sets of orthogonal polynomials) has been subject of active research (see [3], [5], [10]). The aim is often the study of the spaces of the domain of definition of iterated powers (and also powers of the square root) of a given unbounded self-adjoint operator that stems from a differential equation, whose spectral set is a known sequence of orthogonal polynomials.…”