Abstract. We consider the Riemann means of trigonometric series. Results proved: (i) A necessary and sufficient condition for a trigonometric series to be the Fourier series of a function in the periodic real Hardy space $H^1(T)$ is the boundedness of its Riemann means $S_h$ in $H^1$-norm. (ii) A necessary and sufficient condition for a trigonometric series to be the Fourier series of a function in the periodic ${\rm BMO}$ space is the boundedness of its Riemann means in the ${\rm BMO}$-norm.

AMS Subject Classification (1991): 42A16, 42A24, 42A38

Received February 3, 1999, and in revised form March 9, 1999. (Registered under 2704/2009.)