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

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.