PRELIMINARIES AND NOTATION i hoids, for all f E S(R)The set of all such distributions is called S(R).Any polynomially bounded function F can be interpreted as a distribution, with F()dr F(E)f(a). Another example is the so-called" 6-function"of Dirac, &()=f(O). A distribution T is said to be supported in a, b l if T()=0 for all functions f the support of which has empty intersection with [a, b]. One can define the Fourier transform FT or T of a distribution T by T()=T()(if T is a function, then this coincides with our earlier definition). There exists a version of the Paley-Wiener theorem for distributions: an entire function T(S) is the analytic extension of the Fourier transform of a distribution T in S(R) supported in a, bl if and only if, for some NEN, CN>0, T()|≤Cx(1+|) lmIm≥0 ≤0 The only measure we will use is Lebesgue measure on R and R". We will often denote the(Lebesgue)measure of S by SI; in particular, l(a, bl=b-a (where b>a) Well-known theorems from measure and integration theory which we will use include Fatou,s lemma. In20,fn(r)-f(z)almost everywhere(i.e, the get o f points where pomtuLse convergence fauls has zero measure wnth respect to Lebesgue measure), then dr f(=)s limsup dr/n(=) In particular, if thus lim sup is finite, then f is integrable (The lim sup of a sequence is defined by limsup an= lim [sup ak;k2n; every sequence, even if it does not have a limit(such as an =(-1)"),has a lim sup(which may be oo); for sequences that converge to a limit, the lim sup coincides with the limit. Dominated convergence theorem. Suppose fn(r)-f(=)almost every where.Jf|∫n(x)≤g(x) for all n,and∫drg(x)<∞, then f is integrable,, and dxf(=)=lim dr fn(z) Fubini' g theorem.Jf∫dr∫d|f(x,y)<o,ten ∫(x,y) dy dr f(a,y)