Martingale inequalities for spline sequences

We show that D. Lépingle's $L_1\left({\ell }_2\right)$ -inequality $\begin{array}{c}{\left({\left(\sum _n,E,{\left[f_n\right|F_{n-1}]}^2\right)}^{1/2}\right)}_1\le 2·{\left({\left(\sum _n,f_n^2\right)}^{1/2}\right)}_1,f_n\in F_n,\end{array}$ extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that $f_n$ is contained in a suitable spline space $S\left(F_n\right)$ . This is done provided the filtration $\left(F_n\right)$ satisfies a certain regularity condition depending on the degree of smoothness of the functions contained in $S\left(F_n\right)$ . As a by-product, we also obtain a spline version of $H_1$ - $\mathrm{BMO}$ duality under this assumption.