Orthogonal expansion of real polynomials, location of zeros, and an L2 inequality

Let f(z) = a(0)phi (0)(z) + a(1)phi (1)(z) + ... + a(n)phi (n)(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 Turan 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 L-2 inequality giving a lower estimate for the product of two polynomials. (C) 2001 Academic Press.