The representations of functions by orthogonal series possessing martingale properties

R. S. Davtyan

Matematičeskie zametki, Tome 19 (1976) no. 5, pp. 673-680

Résumé

Let $\mathscr F_\infty$ be the minimal $\sigma$-algebra generated by the orthogonal system $\{\varphi_n(x)\}$, defined on the space $(X,S,\mu)$ of finite measure. For a certain class of orthonormal systems one proves that for any $\mathscr F_\infty$-measurable function $f(x)$, which is finite almost everywhere, there exists a series $\sum_{n=1}^\infty a_n\varphi_n(x)$ which converges absolutely to $f(x)$ almost everywhere. This result represents an extension of a theorem by R. Gundy on the representation of functions by orthogonal series possessing martingale properties.