Estimates of Chebyshev polynomials: Inequalities on certain complex domains

Abstract

The Chebyshev polynomials are orthogonal Gegenbauer polynomials that are important in numerical analysis. They can be defined through simple trigonometric identities, a recursion relation and an orthogonalization process. Their properties make them an excellent tool in approximating other functions.

In this paper we introduce different definitions and discuss inequalities and intersections of Chebyshev polynomials of type one and two on the complex domain. Gathering together previous studies, we introduce the different definitions and discuss some extensions onto the complex domain. We give some basic properties and then derive explicit expressions for the zeros and intersection points of different order Chebyshev polynomials of the same type on the real domain, showing that all intersections and zeros occur on the interval [-1,1]. From this we expand to discuss intersection points of the absolute values of different order Chebyshev polynomials on the complex domain, analysing them using some plotted examples. With intuition from observations we then show that the absolute values of lower order Chebyshev polynomials can be greater than or equal to those of higher order Chebyshev polynomials only in a small domain bounded by the unit circle and an ellipse. While doing this we also discuss and rederive a known boundedness property on elliptical domains of the absolute values of Chebyshev polynomials.