The differentiability of Fourier gap series and “Riemann's example” of a continuous, nondifferentiable function

Abstract

We give a general Tauberian gap theorem for a class of Fourier kernels which includes that of the Hankel transform F(x) = ƒ 0 ∞ √ xu J v(xu) ƒ(u) du, v ⩾ − 1 2 . Further, we discuss applications to Fourier gap series and the differentiability of g(x) = ∑ n = 1 ∞ ( sin πn 2x) πn μ , 1 ⩽ μ < 3 , a series supposedly due to Riemann, studied by G. H. Hardy in 1916.