Order of uniform approximation by polynomial interpolation in the complex plane and beyond
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For Lagrange polynomial interpolation on open arcs X=γ in , it is well-known that the Lebesgue constant for the family of Chebyshev points x_n:={x_n,j}^n_j=0 on [-1,1]⊂ has growth order of O(log(n)). The same growth order was shown in <cit.> for the Lebesgue constant of the family z^**_n:={z_n,j^**}^n_j=0 of some properly adjusted Fejér points on a rectifiable smooth open arc γ⊂. On the other hand, in our recent work <cit.>, it was observed that if the smooth open arc γ is replaced by an L-shape arc γ_0 ⊂ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fejér points z_n^*:={z_n,j^*}^n_j=0 on γ, and that the rate of growth for the corresponding Lebesgue constant L_z^*_n is as fast as c log^2(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 γ_0 consisting of two line segments of the same length that meet at the angle of π/2, the growth rate of the Lebesgue constant L_z_n^* is at least as fast as O(Log^2(n)), with limsupL_z_n^*/log^2(n) = ∞; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment z_n^**:={z_n,j^**}^n_j=0 of the Fejér points on γ will be described to assure the growth rate of L_ z_n^** to be exactly O(Log^2(n)).
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