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

Abstract

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arc γ0⊂ℂ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz–Zygmund inequalities are no longer valid for the family of Fejér points zn∗≔{zn,j∗}j=0n on γ, and that the rate of growth for the corresponding Lebesgue constant Lzn∗ is as fast as clog2(n) for some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