On almost everywhere convergence of tensor product spline projections

Let $d\in\mathbb N$ and $f$ be a function in the Orlicz class $L(\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\mathbb{R}^d$. Given partitions $\Delta_1,\ldots,$ $\Delta_d$ of $[0,1]$, we first prove that the orthogonal projection $P_{(\Delta_1,\dots,\Delta_d)}(f)$ onto the space of tensor product splines with arbitrary orders $(k_1,\dots, k_d)$ and knots $\Delta_1,\ldots,\Delta_d$ converges to $f$ almost everywhere as the mesh diameters $|\Delta_1|,\ldots, |\Delta_{d}|$ tend to zero. This extends the one-dimensional result in [Passenbrunner and Shadrin, Journal of Approximation Theory, 2014] to arbitrary dimensions. In a second step, we show that this result is optimal, i.e., given any "bigger" Orlicz class $X=\sigma(L)L(\log^+ L)^{d-1}$ with an arbitrary function $\sigma$ tending to zero at infinity, there exists a function $\varphi\in X$ and partitions of the unit cube such that the orthogonal projections of $\varphi$ do not converge almost everywhere.