ON LAGRANGE INTERPOLATION AT DISTURBED ROOTS OF UNITY

Let Z(nk) = e(it)nk, 0 less-than-or-equal-to t(n0) < ... < t(nn) < 2pi , f a function in the disc algebra A , and mu(n) = max{\t(nk) - 2kpi/(n + 1)\: 0 less-than-or-equal-to k less-than-or-equal-to n} . Denote by L(n)(f; .) the polynomial of degree n that agrees with f at {Z(nk): k = 0, ... , n} In this paper, we prove that for every p, 0 < p < infinity, there exists a delta(p) > 0, such that \\L(n)(f; .) - f\\p = O(omega(f; 1/n)) whenever mu(n) less-than-or-equal-to delta(p)/(n + 1) . It must be emphasized that delta(p) necessarily depends on p , in the sense that there exists a family {z(nk): k = 0, ... , n} with mu(n) = delta2/(n + 1) and such that \\L(n)(f; .) - f\\2 = O(omega(f; 1/n)) for all f is-an-element-of A but sup{\\Ln(f; .)\\p: f is-an-element-of A, \\f\\infinity = 1} diverges for sufficiently large values of p. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for {Z(nk)}.