  We prove the stability under integration and under Fourier transform of a
concrete class of functions containing all globally subanalytic functions and
their complex exponentials. This paper extends the investigation started in
[J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages
analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7
(1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under
integration of sums of products of real globally subanalytic functions and
their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched
framework including oscillatory functions. It provides a new example of
fruitful interaction between analysis and singularity theory.
