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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, general
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