Rapid Decrease of Entire Functions along the Real Axis and the Uniqueness of the Fourier-Transform Pair

We study integral classes of entire functions rapidly decreasing along the real axis, i.e., classes of entire functions whose rapid decrease along the real axis is brought about by the existence of definite weighted integrals. We establish a relationship between these classes and some classes known earlier, in which the decrease of functions along the real axis is given by uniform estimates. As an application, we obtain a concrete realization (also in the integral sense) of the well-known Wiener thesis that the "pair $f$ and $\widehat{f}$ cannot be very small at infinity".