Bounds and Inequalities for General Orthogonal Polynomials on Finite Intervals

✍ Scribed by Y.G. Shi

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 732 KB
• Volume: 73
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1044

✦ Synopsis

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 (L^{2}) version of the principle of contamination. The main results are: we obtain farreaching generalizations of the important results of (\mathrm{P}). Nevai on divergence of Lagrange interpolation in (L^{p}) 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. Theory 43 (1985), Theorem, p. 190] and give new answers to Problems VIII and IX of P. Turán [J. Approx. Theory 29 (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. Theory 29 (1980), p. 71]. 1993 Academic Press. Inc.