Interpolatory properties of best L2-approximants

AbstractLet ƒbe a continuous function and sn be the polynomial of degree at most n of best L2(μ)-approximation to ƒon [-1,1]. Let Zn(ƒ):=\s{xϵ[-1,1]:ƒ(x)−sn(x) = 0\s}. Under mild conditions on the measure μ, we prove that ∪ Zn(ƒ) is dense in [-1,1]. This answers a question posed independently by A. Kroó and V. Tikhomiroff. It also provides an analogue of the results of Kadec and Tashev (for L∞) and Kroó and Peherstorfer (for L1) for least squares approximation