Mellin transform in weak functions and Müntz formula (Q1411419)

Starting with the Müntz formula [\textit{C. H. Müntz}, Mat. Tidsskr. B 1922, 39--47 (1922; JFM 48.0346.02)] \[ \zeta(s) \int^\infty_0 y^{s-1} f(y)\,dy= \int^\infty_0 x^{s-1} \left\{\sum^\infty_{k=1} f(kx)-\frac 1x \int^\infty_0 f(v) \,dv\right\} \,dx\tag{1} \] where \(s=\sigma+it\), \(0< \sigma<1\), \(\zeta(s)\) denotes the Riemann zeta-function, \(f(x)\), \(f'(x)\) are continuous in any finite interval \([0,A)\) and \(O(x^{-\alpha})\), \(O (x^{-\beta})\), respectively, as \(x\mapsto \infty\), \(1<\alpha\), \(1< \beta \), the authors derive the following analogue for the Hurwitz zeta-function (generalized Müntz formula) \[ \zeta(s,a) \int^\infty_0 x^{s-1}f(x)\, dx =\int^\infty_0 x^{s-1} \left[ \sum^\infty_{k=0} f\bigl((k+a) x\bigr)- \frac 1x \int^\infty_0 f(v)\,dv\right]\,dx,\;0<\sigma<1. \] Finally they generalized the modified Müntz formula to generalized functions. \{Note: The authors use throughout their paper the wrong surname Münts.\}