On Almost Everywhere Convergence of Tensor Product Spline Projections

Abstract

Let $d\in \mathbb{N}$, and let $f$ 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 knot sequences ${\Delta }_1,\dots ,{\Delta }_d$ on $[0,1]$, we first prove that the orthogonal projection $P_{({\Delta }_1,\dots ,{\Delta }_d)}\left(f\right)$ onto the space of tensor product splines with arbitrary orders $(k_1,\dots ,k_d)$ and knots ${\Delta }_1,\dots ,{\Delta }_d$ converges to $f$ almost everywhere as the mesh diameters $\left|{\Delta }_1\right|,\dots ,\left|{\Delta }_d\right|$ tend to zero. This extends the one-dimensional result in [9] to arbitrary dimensions.

In the second step, we show that this result is optimal, that is, given any “bigger” Orlicz class $X=\sigma \left(L\right)L({log}^{+}L{)}^{d-1}$ with an arbitrary function $\sigma$ tending to zero at infinity, there exist a function $\phi \in X$ and partitions of the unit cube such that the orthogonal projections of $\phi$ do not converge almost everywhere.