QUADRATURE-FORMULAS AND POLYNOMIAL INEQUALITIES

In this paper we prove several inequalities for polynomials and trigon ometric polynomials. They are all obtained as applications of certain quadrature formulae, some of which are proved here for the first time. Such an application of a Gaussian quadrature formula was pointed out by Bojanov in 1986 (see East. J. Approx. 1 (1995), 37-46; J. Approx. T heory, 83 (1995), 175-181). Coincidentally, in the same year, it was s hown how an inequality for entire functions of exponential type belong ing to L-2(R) could be deduced from a Gaussian quadrature formula for the doubly infinite integral integral(-infinity)(infinity) f(x) dx. (C ) 1995 Academic Press.