Left-definite theory with applications to orthogonal polynomials
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摘要
In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {Hr}r>0 of Hilbert spaces and a continuum of {Ar}r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.
论文关键词:primary,34B30,47B25,47B65,secondary,33C65,34B20,Self-adjoint operator,Hilbert space,Sobolev space,Dirichlet inner product,Left-definite Hilbert space,Left-definite self-adjoint operator,Laguerre polynomials,Stirling numbers of the second kind,Legendre–Stirling numbers,Jacobi–Stirling numbers
论文评审过程:Received 19 February 2008, Available online 26 February 2009.
论文官网地址:https://doi.org/10.1016/j.cam.2009.02.058