Bounding partial sums of Fourier series in weighted L2-norms, with applications to matrix analysis

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 Szegö (Ann. Mat. Pura Appl. 51 (1960) 107), we present explicit upper bounds for supn||Sn|| in terms of w. thors 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.