Interpolation of functions by the least squares method (Q1569290)

Let \(m,n\in\mathbb{N}\) with \(m> 2n\) be given. For a \(2\pi\)-periodic continuous function \(f\), let \(T_n(f)\) be the trigonometric polynomial (of order at most \(n\)) of least squares approximation at the nodes \(x_k= 2\pi k/m\) \((k= 0,1,\dots, m-1)\). Then the norm of the operator \(T_n: C\to C\) is equal to the maximum norm (denoted by \(L_{2n+1}(m)\)) of the function \[ L_{2n+1}(m, x)= {1\over m} \sum^{m-1}_{k=0} \Biggl|{\sin{2n+1\over 2} (x_k- x)\over \sin{1\over 2}(x_k- x)}\Biggr|. \] The author studies the asymptotic behaviour of \(L_n(m)\).