Orthogonal projectors onto spaces of periodic splines

The main result of this paper is a proof that for any integrable function f on the torus, any sequence of its orthogonal projections (P˜nf) onto periodic spline spaces with arbitrary knots Δ˜n and arbitrary polynomial degree converges to f almost everywhere with respect to the Lebesgue measure, provided the mesh diameter ∣Δ˜n∣ tends to zero. We also give a new and simpler proof of the fact that the operators P˜n are bounded on L∞ independently of the knots Δ˜n.