On best \(p\)-approximation from affine subspaces: Asymptotic expansion (Q1608746)

Let \(A\) be an affine subspace of \(\mathbb{R}^n\), and let \(h \in \mathbb{R}^n\). Let \(h_p\) represent the (unique) best \(l_p(n)\)-approximation to \(h\) from \(A\). The best \(l_{\infty}(n)\)-approximation to \(h\) from \(A\) is not necessarily unique, but the Pólya algorithm is that, as \(p\to \infty\), \(h_p\) converges to a best \(l_{\infty}(n)\)-approximation to \(h\) from \(A\), say \(h_{\infty}^*\). It is also known that the rate of convergence is no worse then \(1\over p\). This work extends that classical result on the rate of convergence by showing that for any integer \(m\), there are \(a_1, a_2, \dots , a_m \in \mathbb{R}^n\) and a \(\gamma_p^m \in \mathbb{R}^n\) such that, \[ h_p = h_{\infty}^* + { {a_1}\over{p-1} } + { {a_2}\over{(p-1)^2} } + \cdots + { {a_m}\over{(p-1)^m} } + \gamma_p^m, \] where \(\|\gamma_p^m\|=O({ 1\over{p^{m+1}} })\).