NOTES ON STEKLOVS CONJECTURE IN L-P AND ON DIVERGENCE OF LAGRANGE INTERPOLATION IN L-P

Given a compact interval Delta, it is shown that for E. A. Rakhmanov's weight w on Delta which is bounded from below by the Chebyshev weight nu on Delta (1982, Math. USSR Sb. 42, 263) the corresponding orthonor mal polynomials are unbounded in every L-nu(p) (and L-w(p)) with p > 2 and also that the Lagrange interpolation process Based on their zeros diverges in every L-nu(p) with p > 2 for some continuous f. This yiel ds an affirmative answer to Conjecture 2.9 in ''Research Problems in O rthogonal Polynomials'' (1989, in ''Approximation Theory, VI,'' Vol. 2 , p. 454; (C. K. Chui, L. L. Schumaker, and J. D. Ward, Eds.), Academi c Press, New York) a positive answer to Problem 8, and a negative answ er to Problem 10 of P. Turan (1980, J. Approx. Theory 29, 32-33). (C) 1997 Academic Press.