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Eigenvalues, almost periodic functions, and the derivative of an integral. (English) Zbl 1126.26010

The authors prove that if a function \(g\) is nonconstant and almost periodic on \(\mathbb R\), then the limit \(\lim_{z\to \infty}\int_0^{\infty}g(y+z)e^{-y}\,dy\) does not exist. Using this result they show that every nonconstant almost periodic function \(g\) has the property that \(G(x)=\int_0^xg(-\log t)\,dt\) is not differentiable from the right at \(0\).

MSC:

26A36 Antidifferentiation
26A46 Absolutely continuous real functions in one variable
42A75 Classical almost periodic functions, mean periodic functions
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