On almost everywhere convergence of orthogonal spline projections with arbitrary knots

The main result of this paper is a proof that, for any f∈L1[a,b]f∈L1[a,b], a sequence of its orthogonal projections (PΔn(f))(PΔn(f)) onto splines of order kk with arbitrary knots ΔnΔn converges almost everywhere provided that the mesh diameter |Δn||Δn| tends to zero, namely f∈L1[a,b]⇒PΔn(f,x)→f(x)a.e.(|Δn|→0). This extends the earlier result that, for f∈Lpf∈Lp, we have convergence PΔn(f)→fPΔn(f)→f in the LpLp-norm for 1≤p≤∞1≤p≤∞, where we interpret L∞L∞ as the space of continuous functions.