On Polynomial Approximation of Square Integrable Functions on a Subarc of the Unit Circle

Let E be an arc on the unit circle and let L²(E) be the space of all square integrable functions on E. Using the Banach–Steinhaus Theorem and the weak* compactness of the unit ball in the Hardy space, we study the L² approximation of functions in L²(E) by polynomials. In particular, we will investigate the size of the L² norms of the approximating polynomials in the complementary arc E of E. The key theme of this work is to highlight the fact that the benefit of achieving good approximation for a function over the arc E by polynomials is more than offset by the large norms of such approximating polynomials on the complementary arc E.