Notes on Steklovʹs Conjecture inLpand on Divergence of Lagrange Interpolation inLp

Given a compact intervalΔ, it is shown that for E. A. Rakhmanovʹs weightwonΔwhich is bounded from below by the Chebyshev weightvonΔ(1982,Math. USSR Sb.42, 263) the corresponding orthonormal polynomials are unbounded in everyLpv(andLpw) withp>2 and also that the Lagrange interpolation process based on their zeros diverges in everyLpvwithp>2 for some continuousf. This yields an affirmative answer to Conjecture 2.9 in“Research Problems in Orthogonal Polynomials” (1989,in“Approximation Theory, VI,” Vol. 2, p. 454; (C. K. Chui, L. L. Schumaker, and J. D. Ward, Eds.), Academic Press, New York) a positive answer to Problem 8, and a negative answer to Problem 10 of P. Turán (1980,J. Approx. Theory29, 32–33).