On almost everywhere convergence of orthogonal spline projections with arbitrary knots

The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{Δ_n}(f))$ onto splines of order $k$ with arbitrary knots $Δ_n$, converges almost everywhere provided that the mesh diameter $|Δ_n|$ 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 $P_{Δ_n}(f) \to f$ in the $L_p$-norm for $1 \le p \le \infty$.}

No reviews yet.

No comments yet.