On the integrals of Riemann-Lebesgue type (Q1384446)

The following theorem is proved. Let \(f:[0,\delta]\to\mathbb{C}\) and \(g:[0,\infty)\to \mathbb{C}\) be measurable functions, \(f=f_1+f_2\). If the following conditions are satisfied for some \(p\geq 0\) and \(\delta> 0\): \hskip 7mm (i) \(\lim_{t\to+0} tf_1(t)= a\), \hskip 7mm (ii) \(t^{p+ 1}f_1(t)\) is of bounded variation on \([0,\delta]\) and \[ V(t^{p+ 1}f_1(t); 0,h)= O(h^p)\quad\text{as}\quad h\to+ 0, \] \hskip 7mm (iii) \(\lim_{\lambda\to\infty} \int^\delta_0 f_2(t)g(\lambda t)dt= 0\), \hskip 7mm (iv) \(t^{-1}g(t)\) is locally integrable on \([0,\infty)\) and the improper integral \hskip 14mm \(\int^\infty_0 t^{-1}g(t)dt\) converges, \hskip 7mm (v) \(\sup_{t\geq 0}\left|\int^t_0 g(s)ds\right|< \infty\), then \[ \lim_{\lambda\to\infty} \int^\delta_0 f(t)g(\lambda t)dt= a\int^\infty_0 {g(t)\over t} dt. \] The generalized Young test, Dirichlet-Jordan test, de la Vallée-Poussin test, and Dini test are corollaries of the above theorem.