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## Hyperbolicity preserving differential operators and classifications of orthogonal multiplier sequences

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### Item Summary

Title: | Hyperbolicity preserving differential operators and classifications of orthogonal multiplier sequences |

Authors: | Bates, Robert D. |

Keywords: | orthogonal multiplier sequences hyperbolicity |

Issue Date: | Dec 2014 |

Publisher: | [Honolulu] : [University of Hawaii at Manoa], [December 2014] |

Abstract: | It is well known that every linear operator, T : R[x] ! R[x], can be represented in the form, T := 1Xk=0 Qk(x)Dk; D := d dx ; where fQk(x)g1k=0 is a sequence of real polynomials. An outstanding open problem in the theory of distribution of zeros of polynomials is to characterize the sequence, fQk(x)g1k =0, such that the operator T is hyperbolicity preserving; i.e., T maps polynomials with only real zeros to polynomials of the same kind. A large portion of this dissertation focuses on diagonal differential operators; that is, T[Bn(x)] := 1Xk=0 Qk(x)Dk!Bn(x) = nBn(x); n 2 N0; where fBn(x)g1n=0 is a simple sequence of real polynomial eigenvectors and f ng1n=0 is the corresponding sequence of real eigenvalues. Our analysis leads to new relations between the eigenvector sequence and the eigenvalue sequence in a diagonal differential operator. In particular, we develop new methods for determining the polynomial coefficients, fQk(x)g1k=0, in cases of Hermite, Laguerre, or monomial linear transformations. We establish a new representation of linear operators and demonstrate novel hyperbolicity properties for this representation. We show that every Hermite or Laguerre multiplier sequence can be expressed as a sum of classical multiplier sequences. Using the Malo-Schur-Szeg}o Composition Theorem, we present a new proof of J. Borcea and P. Brändén's seminal result on the hyperbolicity preservation of finite order differential operators. The hyperbolicity preservation of order two differential operators is studied in minute detail; this leads to a new Turán-Wronskian inequality. In addition, a new algebraic characterization is given for the class of Hermite multiplier sequences. Moreover, we prove that in the case of a Hermite diagonal differential operator, T, the zeros of Qk(x) and Qk+1(x) are interlacing, k 0. We generalize several results of D. Bleecker, G. Csordas, T. Forgács, and A. Piotrowski, partially answer a number of general open problems of T. Craven, G. Csordas, and S. Fisk, and solve a question of M. Chasse. This dissertation concludes with an outline of possible future research and a list of open questions. |

Description: | Ph.D. University of Hawaii at Manoa 2014. Includes bibliographical references. |

URI/DOI: | http://hdl.handle.net/10125/101054 |

Rights: | All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner. |

Appears in Collections: | Ph.D. - Mathematics |

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