On a uniform asymptotic expansion of a Fourier-type integral

An alternative derivation is given for a uniform asymptotic expansion of the integral \[ I\left ( h \right ) = \int _q^z {g\left ( {\sqrt {{y^2} - {q^2}} } \right )\sin yh dy \qquad \left ( {h \to + \infty } \right )} \], which has been obtained recently by Schmidt. Here $q$ varies in the interval $\left [ {0,{q_0}} \right ]$, ${q_0}$ is some fixed number less than $z$, and $z$ is finite. A similar result is obtained for integrals having an infinite range of integration. Realistic bounds are also provided for the error terms associated with the expansions. Our approach is based on a summability method introduced by Olver.