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 \[ f \in L_1[a,b] \Rightarrow P_{Δ_n}(f,x) \to f(x) \quad \mbox{a.e.} \quad (|Δ_n|\to 0)\,. \] 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 $1\le p\le \infty$.}

13 pages

References in corpus (1)

• Unconditionality of orthogonal spline systems in $L^p$

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