Abstract :
In this paper using a new effective approach we deduce some bounds and inequalities for general orthogonal polynomials on finite intervals and give their applications to convergence of orthogonal Fourier series, Lagrange interpolation, orthogonal series with gaps, and Hermite-Fejér interpolation, as well as to the L2 version of the principle of contamination. The main results are: we obtain far-reaching generalizations of the important results of P. Nevai on divergence of Lagrange interpolation in Lp with p < 2 ["Orthogonal Polynomials," Memoirs of the Amer. Math. Soc., Vol. 213, Amer. Math. Soc., Providence, RI, 1979, Corollary 10.18, p. 181; J Approx. Theory43 (1985), Theorem, p. 190] and give new answers to Problems VIII and IX of P. Turán [J. Approx. Theory29 (1980), pp. 32-33]; we extend Turán′s Inequality [Anal. Math.1 (1975), 297-311, Lemma II] to "arbitrary" measures supported in [− 1, 1] and solve Problem LXXI of P. Turán [J. Approx. Theory29 (1980), p. 71].
