Orthogonal Expansion of Real Polynomials, Location of Zeros, and an L2 Inequality

AbstractLet f(z)=a0φ0(z)+a1φ1(z)+…+anφn(z) be a polynomial of degree n, given as an orthogonal expansion with real coefficients. We study the location of the zeros of f relative to an interval and in terms of some of the coefficients. Our main theorem generalizes or refines results due to Turán and Specht. In particular, it includes a best possible criterion for the occurrence of real zeros. Our approach also allows us to establish a weighted L2 inequality giving a lower estimate for the product of two polynomials