Abstract
In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.
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.