On almost everywhere convergence of orthogonal spline projections with arbitrary knots

TL;DR

This paper proves that orthogonal spline projections with arbitrary knots converge almost everywhere for any integrable function as the mesh size shrinks, extending previous norm convergence results.

Contribution

It establishes almost everywhere convergence of spline projections with arbitrary knots for all $L_1$ functions, generalizing earlier $L_p$-norm convergence results.

Findings

Almost everywhere convergence for $L_1$ functions.

Extension of previous $L_p$-norm convergence results.

Applicable to splines of arbitrary knots and order.

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 $1 \le p \le \infty$.}