Left-definite theory with applications to orthogonal polynomials |
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Authors: | Andrea Bruder Davut Tuncer |
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Institution: | a Department of Mathematics, Baylor University, One Bear Place #97328 Waco, TX 76798-7328, USA b Department of Mathematics, Westminster College, Foster Hall, Salt Lake City, UT 84105, USA |
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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 {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. |
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Keywords: | primary 34B30 47B25 47B65 secondary 33C65 34B20 |
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