On Almost Everywhere Convergence of Tensor Product Spline Projections

Let d∈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 Rd. Given knot sequences Δ1,…,Δd on [0,1], we first prove that the orthogonal projection P(Δ1,…,Δd)(f) onto the space of tensor product splines with arbitrary orders (k1,…,kd) and knots Δ1,…,Δd converges to f almost everywhere as the mesh diameters |Δ1|,…,|Δd| 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=σ(L)L(log+L)d−1 with an arbitrary function σ tending to zero at infinity, there exist a function φ∈X and partitions of the unit cube such that the orthogonal projections of φ do not converge almost everywhere.