Abstract
Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of a collection of convex cyclic $n$-gons whose vertices fill the whole unit circle $\mathbb T$. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of $n$. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.