On the convergence and Gibbs phenomenon of Franklin series (Q1362396)

\textit{Z. Ciesielski} [Stud. Math. 27, 289-323 (1966; Zbl 0148.04702)] proved that if \(f\in L^1[0,1]\) and the derivative of the indefinite integral of \(f\) equals \(f(t_0)\) at some point \(t_0\in[0,1]\), then the Franklin-Fourier series of \(f\) converges (to \(f\)) at \(t_0\). Here, the author obtains pointwise estimates of the \(\limsup\) and \(\liminf\) of the Franklin-Fourier-Stieltjes series \(S\) of a Riemann integrable function \(g\) at \(t_0\) in terms of upper and lower derivates of \(g\) at \(t_0\) which he uses to show that if \(g'(t_0)\) exists then \(S_n(t)\) converges to \(g'(t_0)\) uniformly at \(t_0\). A two-dimensional version of this result is obtained for bounded convergence, i.e., the \(\limsup\), \(\liminf\), and limit of \(S_{mn}\) are taken with \((m,n)\) restricted to a positive cone in the first quadrant. The author also examines Gibbs phenomenon for Franklin-Fourier series. Unlike the classical trigonometric case where the Gibbs constant is approximately 1.7, the Gibbs ``constant'' for Franklin-Fourier series varies between \[ 1+{\sqrt 3-1\over 3}\leq G(t_0)\leq 1+{4\sqrt 3\over 3(3+\sqrt 3)}. \] It is, however, almost everywhere constant with value equal the upper limit of the range above.