Martingale inequalities for spline sequences

We show that D. Lépingle’s [Formula: see text] -inequality [Formula: see text] extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that [Formula: see text] is contained in a suitable spline space [Formula: see text] . This is done provided the filtration [Formula: see text] satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in [Formula: see text] . As a by-product, we also obtain a spline version of [Formula: see text] -[Formula: see text] duality under this assumption.