Order of uniform approximation by polynomial interpolation in the complex plane and beyond

Let X denote a compact set in the complex plane ℂ and zn≔{zn,j}j=0n be family of points that lie on X. Then norm ‖Azn‖ operator Azn maps Banach space C(X) to Πn polynomials degree ≤n, defined by Aznf=pn, where pn∈Πn is polynomial interpolates f∈C(X) at zn, called Lebesgue constant zn. While bulk this paper study Lagrange interpolation Fejér zn∗≔{zn,j∗}j=0n, their proper adjustment, an open arc X=γ which does not cross itself, general spirit our presentation carries over other settings representations functions simple closed piece-wise smooth curve X=Γ, boundary Jordan domain D⊂ℂ; instead interpolation, replaced linear AFn, (generalized) Hardy Hp(D), for p≥1, analytic f D with non-tangential limit f∗∈Lp(Γ), nth partial sum Sn(⋅;f) Faber series representation D. The importance ‖AFn‖ if Pn∗∈Πn best approximant approximation error ϵn(f;Γ)≔‖f−Pn∗(⋅;f)‖Hp, then can used replace Pn∗ ‖f−Sn(⋅;f)‖Hp≤(1+‖AFn‖)ϵn(f;Γ). This consideration, along discussion related problems results inspired work Prof. J. Korevaar, will presented final section present paper. For arcs ℂ, it well-known Chebyshev xn≔{xn,j}j=0n [−1,1]⊂R has growth order O(log(n)). same was shown Zhong Zhu (1995) zn∗∗≔{zn,j∗∗}j=0n some properly adjusted rectifiable γ⊂ℂ. On hand, recent Chui (2021), observed γ L-shape γ0⊂ℂ consisting two line segments, numerical experiments suggest Marcinkiewicz–Zygmund inequalities are no longer valid zn∗≔{zn,j∗}j=0n γ, rate corresponding Lzn∗ as fast clog2(n) c>0. main objective 3-fold: firstly, special case γ0 segments length meet angle π/2, least O(log2(n)), limsupLzn∗log2(n)=∞; secondly, (modified) fail hold; thirdly, adjustment described assure Lzn∗∗ exactly O(log2(n)).