Unbounded Widom factors for orthogonal and residual polynomials

Official URL: https://dx.doi.org/10.15672/hujms.1786276

We consider Widom factors for two extremal problems on compact sets K ⊂ ℝ: the L2 Widom factors W2,n(µK) of monic orthogonal polynomials with respect to the equilibrium measure µK, and the residual Widom factors W(x0) ∞,n (K) for residual polynomials normalized at an exterior point x0 ∉ K. Using weakly equilibrium Cantor sets K(γ), we prove that for any sequence (cn) with subexponential growth there exists a non-polar K(γ) such that W2,n(µK(γ)) ≥ cn for all n. Moreover, for each x0 ∉ ℝ \ K(γ) we obtain power-type lower bounds of the form (Formula presented) n with τx0 ∉ (0, 1), holding for all n in unbounded gaps and along a subsequence in bounded gaps.