Let , and let be a function in the Orlicz class defined on the unit cube in . Given knot sequences on , we first prove that the orthogonal projection onto the space of tensor product splines with arbitrary orders and knots converges to almost everywhere as the mesh diameters tend to zero. This extends the one-dimensional result in  to arbitrary dimensions.
In the second step, we show that this result is optimal, that is, given any “bigger” Orlicz class with an arbitrary function tending to zero at infinity, there exist a function and partitions of the unit cube such that the orthogonal projections of do not converge almost everywhere.
Markus Passenbrunner. Joscha Prochno. "On Almost Everywhere Convergence of Tensor Product Spline Projections." Michigan Math. J. 68 (1) 3 - 17, April 2019. https://doi.org/10.1307/mmj/1541667630