Abstract
We investigate the relationship between the Bergman shift operator and the support of the corresponding measure. We pay special attention to the situation when the measure of orthogonality is concentrated on a polynomial lemniscate. As an application of our new results, we obtain a ratio asymptotic result for a wide variety of measures supported on polynomial lemniscates.
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Acknowledgments
It is a pleasure to thank Ed Saff for encouraging me to pursue this line of investigation and for much useful discussion. I would also like to thank Barry Simon for useful feedback concerning this work.
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Communicated by Doron S. Lubinsky.
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Simanek, B. The Bergman Shift Operator on Polynomial Lemniscates. Constr Approx 41, 113–131 (2015). https://doi.org/10.1007/s00365-014-9265-z
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DOI: https://doi.org/10.1007/s00365-014-9265-z
Keywords
- Weak asymptotic measures
- Ratio asymptotics
- Bergman Polynomials
- Bergman shift operator
- Hessenberg matrices
- Christoffel transform