Interpolatory properties of best L2-approximants

Let ƒbe a continuous function and s_n be the polynomial of degree at mostn of best L₂(μ)-approximation to ƒon -1,1]. Let Z_n(ƒ):=\s{xε-1,1]:ƒ(x)−s_n(x) = 0\s}. Under mild conditions on the measure μ, we prove that Z_n(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.