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 ${\tilde{\mathrm{\Delta }}}_n$ and arbitrary polynomial degree converges to $f$ almost everywhere with respect to the Lebesgue measure, provided the mesh diameter $\mathrm{\mid }{\tilde{\mathrm{\Delta }}}_n\mathrm{\mid }$ tends to zero. We also give a proof of the fact that the operators ${\tilde{P}}_n$ are bounded on $L^{\mathrm{\infty }}$ independently of the knots ${\tilde{\mathrm{\Delta }}}_n$.