On almost everywhere convergence of orthogonal spline projections with arbitrary knots

Abstract

The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{\Delta_n}(f))$ onto splines of order $k$ with arbitrary knots $\Delta_n$, converges almost everywhere provided that the mesh diameter $|\Delta_n|$ tends to zero, namely \[ f \in L_1[a,b] \Rightarrow P_{\Delta_n}(f,x) \to f(x) \quad \mbox{a.e.} \quad (|\Delta_n|\to 0)\,. \] This extends the earlier result that, for $f \in L_p$, we have convergence $P_{\Delta_n}(f) \to f$ in the $L_p$-norm for $1 \le p \le \infty$.}