Let \(f(x)\) be a periodic function with period \(2\pi\) and \(L\)-integrable over \((-\pi, \pi)\). In 1926 Hardy and Littlewood considered the Cesàro summability of the series \[ \sum [s_n(x) - s]/n, \tag{*} \] where \(s_n(x)\) denotes the \(n\)th partial sum of the Fourier series and \(s\) an appropriate number, and to which they termed as associated Fourier series. Subsequently A. Zygmund [Trigonometrical series. New York: Dover Publications (1955; Zbl 0065.05604), cf. also reviews of the first ed. (1935; Zbl 0011.01703; JFM 61.0263.03) and 2nd ed. (1959; Zbl 0085.05601)] gave a necessary and sufficient condition for the convergence of the series (*). In this note the author obtains a sufficient condition for its harmonic summability. He proves the following result:
Let \(\varphi(t) = \{f(x + t) + f(x - t) - 2s\}\) and \(\varphi_1(t) = \tfrac1{t} \int_0^t \varphi(u)\,du\). If \[ \int_0^t \vert\varphi_1(u)\vert\,du = o(t/\log\tfrac1{t})\quad\text{and}\quad \frac1{2\pi} \int_0^\pi \varphi(t)\operatorname{cosec} \tfrac12 t \,dt \] exists, then the series (*) is harmonic summable.