On the Hausdorff moment problem (Q5942581)

Let \(\omega_i(x)(1+x)^{-1}\in L_1(R^1_+)\), \(i=1,2\) and \(\omega_i(x) \geq 0\). The author describes a solution of the Hausdorff moment problem. The following integral equation of the first kind \[ \int_0^\infty e^{-st}d\alpha (t)= \left(\int_0^\infty \frac{\omega_1 (x)}{s+x} dx\right) \left(\int_0^\infty \frac{\omega_2 (x)}{s+x} dx\right)^{-1}, \qquad s>0, \tag{1} \] is considered under the assumptions: i) \(\omega_1(x+\Delta)\omega_2(x-\Delta)\leq \omega_1(x-\Delta)\omega_2(x+\Delta)\) for any \(x, \Delta>0\) such that \(0<x-\Delta<x<x+\Delta<\infty\); ii) \(\min_{x\in[a,b]}\omega_2 (x)\geq \rho>0\) for a segment \([a,b]\subset (0,\infty)\). He shows that the equation (1) has a non-decreasing solution \(\alpha (x)\) such that \(\alpha (0)=0\) and \[ \alpha (x)\leq \left(\int_0^\infty e^{-xt}\omega_1 (t) dt\right) \left(\int_0^\infty e^{-xt}\omega_2 (t) dt\right)^{-1}, \qquad x>0. \]