On the absolute convergence of trigonometrical series
Tamotsu Tsuchikura, Shigeki Yano · Proceedings of the American Mathematical Society · 1950
EiPn < .0 We shall see in Theorem 2 that the above theorem is no longer true if p/q is-replaced by p'/q' with an even q' and p'$O, or by an irrational number. PROOF OF THEOREM 1. If we put xi-xo = pr/q _ h, then by Fatou' s theorem' (1) E Pn I cos (n(x sk) -an) I < 00 for every odd s. Hence from the identity sin (nx, an) = cos nsh sin (n(x, sh)an) + sin nsh cos (n(x, sh) an) we deduce immediately that (2) E Pn I cos nsh sin (n(x, sh)) an) < 0. Received by the editors October 18, 1948 and, in revised form, May 14, 1949. I See, for example, A. Zygmund, Trigonometrical series, p. 134.