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For integrable functions f defined on the interval [−π,π], we denote the partial sums of the corresponding Fourier series by Sn(f) (n=0,1,2,…). In this paper, we deal with the problem of bounding supn||Sn||, where ||·|| denotes an operator norm induced by a weighted L2-norm for functions f on [−π,π]. For weight functions w belonging to a class introduced by Helson and Szego¨ (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for supn||Sn|| in terms of w.The authors were led to the problem of deriving explicit upper bounds for supn||Sn||, by the need for such bounds in an analysis related to the Kreiss matrix theorem—a famous result in the fields of linear algebra and numerical analysis. Accordingly, the present paper highlights the relevance of bounds on supn||Sn|| to these fields. [Copyright &y& Elsevier]