Martingale inequalities for spline sequences

We show that D. Lépingle's L1(ℓ2) -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 fn is contained in a suitable spline space S(Fn) . This is done provided the filtration (Fn) satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in S(Fn) . As a by-product, we also obtain a spline version of H1 - BMO duality under this assumption.