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