## The differentiability of Fourier gap series and “Riemann’s example” of a continuous, nondifferentiable function.(English)Zbl 0626.42008

Author’s abstract: We give a general Tauberian gap theorem for a class of Fourier kernels which includes that of the Hankel transform $$F(x)=\int^{\infty}_{0}\sqrt{xu}J_{\nu}(xu)f(u)du,\nu\geq -$$. Further, we discuss applications to Fourier gap series and the differentiabiliy of $$g(x)=\sum^{\infty}_{n=1}(\sin \pi n^ 2x)/\pi n^{\mu},$$ $$1\leq \mu <3$$, a series supposedly due to Riemann, studied by G. H. Hardy in 1916.
Reviewer: B.P.Duggal

### MSC:

 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 40E05 Tauberian theorems
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### References:

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