PRELIMINARIES AND NOTATION Inversion of the Fourier transform is then given by ∫(r) V2n/de(F)()=(f)y(x), (0.03 (x)=9(-x) Strictly speaking, (0.0.1),(0.0. 3) are well defined only if f, respectively Ff, are bsolutely integrable; for general L-functions f, e.g. we should define Ff via a limiting process(see also below). We will implicitly assume that the adequate limiting process is used in all cases, and write, with a convenient abuse of no- tation, formulas similar to(0.0. 1)and (0.0.3)even when a limiting process is understood A standard property of the Fourier transform is =(i)4()() drfo()2<∞…/ dE 1512 If()2<∞ with the notation f(e dzf If unction f is compactly supported, i.e. f(=)=0 if r a or I>6, where -oo< a<6<oo. then its Fourier transform f(E)is well defined also for complex 5, and 1(E) s(2m)-1/ dr e(lme)r 1f(z) eb(m)ifIm≥0 (Im E) ifIm5≤0 If f is moreover infinitely differentiable, then the same argument can be applied to f(e), leading to bounds on IEI If(E). For a Coo function f with support a, bl there exist therefore constants CN so that the analytic extension of the Fourier transform of f satisfies (0.0.4) Conversely, any entire function which satisfies bounds of the type(0.0.4)for all NN is the analytic extension of the fourier transform of a Coo function with support in (a, b]. This is the Paley-Wiener theorem We will occasionally encounter (tempered)distributions. These are linear naps T from the set S(R)(consisting of all Coo functions that decay faster than any negative power (1+a)-)to C, such that for all m, n E N, there exists n m for which T()≤ +1fm()