A previous paper of Bleistein, Handelsman and Lew describes the asymptotic behavior of \[F(\omega ) = \mathop {\lim }\limits_{u \to + \infty } \int_0^u {\exp (i\omega t)f(t)dt} \]for certain functions f on $[0, + \infty )$. It estimates the growth or decay of F near $ + \infty $ when f has a suitable asymptotic expansion, then establishes the decay in particular of F near $ + \infty $ when f has certain qualitative properties. This note fills a gap in the preceding work; it gives a qualitative estimate for F when this transform does not decay near $ + \infty $.