Abstract
This paper consists of three parts. In the first part we complete one of the deepest theorems due to Littlewood and Paley [3, 4].1 Our theorem was already proved by Marcinkiewicz and Zygmund [4]. But the proof given here is very simple and direct. In the second part a strong summability theorem is proved. It is the completion of our former theorem [5]. We use the method due to Zygmund [8] for the proof. Finally we prove a strong summability theorem concerning lacunary sequences of partial sums (Zalcwasser [6]). 1. Letf(6) be an integrable function with period 2ir and its Fourier series be f(I)'f= (an cos nO+b,, sin nO), assuming ao=O for the sake of simplicity. If we put z=pei0 and
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