Projections onto piecewise linear functions (Q1379859)

Let \(\pi=(t_i,i \in\mathbb{Z})\) be a given sequence of simple knots on the real line \(\mathbb{R},I_i =(t_{i-1}, t_1)\), \({\mathfrak L}_\pi\) be the space of continuous real-valued functions on \(\mathbb{R}\) which are linear on each \(I_1 (i\in\mathbb{Z})\), and \({\mathfrak L}^2_\pi ={\mathfrak L}\cap L^2 (\mathbb{R})\). Let \(K_\pi (t,s)\) denote the Dirichlet kernel of the orthogonal projection of \(L^2 (\mathbb{R})\) onto \({\mathfrak L}^2_\pi\). The authors consider the order of approximation a function \(f\in L^p (\mathbb{R})\), by \(P_\pi f\), \[ P_\pi f(t)= \int_\mathbb{R} K_\pi(t,s) f(s)ds, \] and the convergence of \(P_\pi f(t)\) at weak Lebesgue points of \(f\) as diameter \(\pi= \sup\{| I_i|: i\in\mathbb{Z}\}\to 0\).