Markus Passenbrunner, Joscha Prochno,
"On Almost Everywhere Convergence of Tensor Product Spline Projections"
, in Michigan Math. J., Vol. 68, Nummer 1, Seite(n) 3-17, 2019
On Almost Everywhere Convergence of Tensor Product Spline Projections
Sprache des Titels:
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 ?1,?,?d on [0,1], we first prove that the orthogonal projection P(?1,?,?d)(f) onto the space of tensor product splines with arbitrary orders (k1,?,kd) and knots ?1,?,?d converges to f almost everywhere as the mesh diameters |?1|,?,|?d|
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 X=?(L)L(log+L)d?1
with an arbitrary function ? tending to zero at infinity, there exist a function ??X and partitions of the unit cube such that the orthogonal projections of ? do not converge almost everywhere.