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

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