Left-definite theory with applications to orthogonal polynomials

  • Authors:

  • Andrea Bruder;Lance L. Littlejohn;Davut Tuncer;R. Wellman

  • Affiliations:

  • Department of Mathematics, Baylor University, One Bear Place #97328 Waco, TX 76798-7328, USA;Department of Mathematics, Baylor University, One Bear Place #97328 Waco, TX 76798-7328, USA;Department of Mathematics, Baylor University, One Bear Place #97328 Waco, TX 76798-7328, USA;Department of Mathematics, Westminster College, Foster Hall, Salt Lake City, UT 84105, USA

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

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Abstract

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 {H"r}"r""0 of Hilbert spaces and a continuum of {A"r}"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.