On joint summability of Fourier series and conjugate series (Q1194687)

Let \(S_ n^ k(f;x)\) denote the \(n\)th Cesàro mean \((C,k)\) of the Fourier series of a \(2\pi\)-periodic Lebesgue integrable function \(f\) at the point \(x\). Considering pointwise convergence in place of uniform convergence and \(S_ n^ k(f;x)\) in place of \(S_ n^ 1(f;x)\), the author proves an anlogue of the following result [see \textit{A. Zygmund}, Trigonometric series, Vol. I (1959; Zbl 0085.056), p. 122]: If \(S_ n^ 1(f;x)-f(x)=o(1/n)\) uniformly in \(x\), then \(f\equiv\) constant. A similar result is also proved for the conjugate series of the Fourier series of \(f\).