Quadrature Formulae and Polynomial Inequalities Original Research Article

In this paper we prove several inequalities for polynomials and trigonometric 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 (seeEast. J. Approx.1(1995), 37–46;J. Approx. Theory83(1995), 175–181). Coincidentally, in the same year, it was shown how an inequality for entire functions of exponential type belonging toL2(R) could be deduced from a Gaussian quadrature formula for the doubly infinite integral ∫∞−∞ f(x) dx.