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 ${\mathrm{\Delta }}_1,\dots ,$ ${\mathrm{\Delta }}_d$ of $[0,1]$, we first prove that the orthogonal projection $P_{({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_d)}(f)$ onto the space of tensor product splines with arbitrary orders $(k_1,\dots ,k_d)$ and knots ${\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_d$ converges to $f$ almost everywhere as the mesh diameters $\mathrm{\mid }{\mathrm{\Delta }}_1\mathrm{\mid },\dots ,\mathrm{\mid }{\mathrm{\Delta }}_d\mathrm{\mid }$ 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 $\phi \in X$ and partitions of the unit cube such that the orthogonal projections of $\phi$ do not converge almost everywhere.