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 (\tilde P_n f)onto periodic spline spaces with arbitrary knots \Tilde Δ_n and arbitrary polynomial degree converges to f almost everywhere with respect to the Lebesgue measure, provided the mesh diameter ∣\Tilde Δ_n∣ tends to zero. We also give a new and simpler proof of the fact that the operators P˜nare bounded on L^∞ independently of the knots \tilde Δ_n.