Abstract
We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter λ which leads to a linear operator pencil. We show that the little and big −1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.
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Acknowledgements
The authors are very grateful to the referees and the editor for many valuable comments and suggestions which helped to improve the manuscript and also gave us some ideas for further developments. The research of LV is supported in part by a research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
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Communicated by Tom H. Koornwinder.
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Derevyagin, M., Vinet, L. & Zhedanov, A. CMV Matrices and Little and Big −1 Jacobi Polynomials. Constr Approx 36, 513–535 (2012). https://doi.org/10.1007/s00365-012-9164-0
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DOI: https://doi.org/10.1007/s00365-012-9164-0
Keywords
- Orthogonal polynomials on the unit circle
- Little and big −1-Jacobi polynomials
- Delsarte–Genin map and its generalization
- CMV matrix
- Jacobi matrix
- Linear pencils