Scaling limits of Chebyshev polynomials at a hard edge

Abstract: Chebyshev polynomials are important objects in approximation theory, characterized by minimizing the uniform norm among monic polynomials. While their global asymptotic behavior is well understood, much less is known about their local structure. In this talk, we consider Chebyshev polynomials near a hard edge of a compact subset of the real line. After an appropriate rescaling, these polynomials form a normal family. However, unlike orthogonal polynomials at a hard edge, the rescaled Chebyshev polynomials do not necessarily converge to a single limit. Instead, different subsequences may converge to different limit functions, which can all be described in terms of conformal maps from the upper half-plane onto comb domains. Moreover, this behavior is generic: For a generic choice of two intervals, all possible limits arise. 

This is based on joint work with Benjamin Eichinger.