Abstract. Two theorems verifying the necessity of certain conditions, which imply the exact order of magnitude of partial sums and Riesz-means for orthogonal series with strict monotone sequences, are generalized such that the necessity of the same conditions is proved for almost monotone sequences.

AMS Subject Classification (1991): 42C15, 40G99

Received April 25, 2003, and in revised form June 2, 2004. (Registered under 5839/2009.)