Properties of local orthonormal systems Part I: Unconditionality in $$L^p$$, $$1<p<\infty$$

Assume that we are given a filtration $$(\mathcal F_n)$$ on a probability space $$(\Omega,\mathcal F,\mathbb {P})$$ of the form that each $$\mathcal F_n$$ is generated by the partition of one atom of $$\mathcal F_{n-1}$$ into two atoms of $$\mathcal F_n$$ having positive measure. Additionally, assume that we are given a finite-dimensional linear space $$S$$ of $$\mathcal F$$-measurable, bounded functions on $$\Omega$$ so that on each atom $$A$$ of any $$\sigma$$-algebra $$\mathcal F_n$$, all $$L^p$$-norms of functions in $$S$$ are comparable independently of $$n$$ or $$A$$. Denote by $$S_n$$ the space of functions that are given locally, on atoms of $$\mathcal F_n$$, by functions in $$S$$ and by $$P_n$$ the orthoprojector (with respect to the inner product in $$L^2(\Omega)$$) onto $$S_n$$. Since $$S = \operatorname{span}\lbrace \mathbbm 1_\Omega \rbrace$$ satisfies the above assumption and $$P_n$$ is then the conditional expectation $$\mathbb {E}_n$$ with respect to $$\mathcal F_n$$, for such filtrations, martingales $$(\mathbb {E}_n f)$$ are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences $$(P_n - P_{n-1})f$$ form an unconditionally convergent series and are democratic in $$L^p$$ for $$1<p<\infty$$. This implies that those differences form a greedy basis in $$L^p$$-spaces for $$1<p<\infty$$.