On polynomial approximation of square integrable functions on a subarc of the unit circle (Q1604424)

Let \(E\) be an arc on the unit circle \(T:|z|= 1\), \(L^2(E)\) the space of all measurable functions which are square integrable on \(E\) with inner product and the \(L^2\)-norm defined as \[ (f,g)= {1\over 2\pi} \int_E f\overline g|dz|,\quad \|f\|= \sqrt{(f,f)}. \] Using the Banach-Steinhaus theorem and the weak\(^*\) compactness of the unit ball in the Hardy space, the author studies the \(L^2\)-approximation of functions in \(L^2(E)\) by polynomials. In particular, the size of the \(L^2\)-norms of the approximating polynomials in the complementary arc \(\widetilde E\) of \(E\) is investigated. It is shown 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 \(\widetilde E\).