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We develop real Paley-Wiener theorems for classes ${\mathcal S}_\omega$ ofultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bangand Andersen for the Schwartz class. We introduce results of this type for theso-called Gabor transform and give a full characterization in terms of Fourierand Wigner transforms for several variables of a Paley-Wiener theorem in thisgeneral setting, which is new in the literature. We also analyze this type ofresults when the support of the function is not compact using polynomials. Someexamples are given.

Real Paley-Wiener theorems in spaces of ultradifferentiable functions