Abstract. Two theorems are proved extending a well-known Paley theorem for orthonormal systems $\{\varphi_n\} $ on the unit interval $[0,1]$ such that $\varphi_n\in{\rm BMO}$.

AMS Subject Classification (1991): 46E35, 26A15

Received July 9, 1997 and in revised form December 10, 1997. (Registered under 2658/2009.)