4 Fourier TransformFor f ∈ L1, the Fourier transform, denotedˆf is defined asˆf(ω) :=Z∞−∞f(t)e−jωtdtSince f ∈ L1, this is well defined at each ω . In fact,ˆf is uniformly continuous,bounded and decays to 0 as ω → ∞.For f ∈ L1TL2, thenˆf also satisfiesˆf ∈ L2, andkfk2=√2πˆf2For f ∈ L2, the transform must defined in terms of limits. Choose a sequence{fk}∞k=1such that fk∈ L1TL2, and fk→ f in L2. This can be done in manyways, for examplefk(t) =0 if |t| > kf(t) if |t| ≤ kThe Fourier transformsˆfkare well-defined, and are a Cauchy sequence inL2, so that fo r any ǫ > 0 , there is an integ er N, such that f or all j, k > N,ˆfj−ˆfk2< ǫSince L2is complete (a ll Cauchy sequences have limits) there is a uniquefunctionˆf in L2such thatlimk→∞ˆfk−ˆf2= 0This function,ˆf is the Fourier (Plancheral) transform of f.The inverse transformation can also be defined, fo r g ∈ L1,ˇg(ω) :=Z∞−∞g(t) ejωtdtAgain, ˇg is uniformly continuous, bounded and decays to 0 as ω → ∞. Usingthe same limit ideas as above, this can be extended to g ∈ L2.Summarizing, for f, g ∈ L2, the Plancheral (Fourier) transformsˆf, ˆg satisfy461. Bothˆf, ˆg ∈ L22. The inner products are relatedZ∞−∞f(t)g(t)dt =12πZ∞−∞ˆf(ω)¯ˆg(ω)dω =3. (following from above) the nor ms satisfy kf k2=1√2πˆf24. The integral forms of the t ransform and its inverse relate the functionsaslimk→∞Zk−kf(t)e−jωtdt −ˆf(ω)2= 0andlimk→∞Zk−k]ˆf(ω)ejωtdω − f(t)2= 04.1 ReferencesThere are a few references for this that I am familar with:1. page 182-185 in Partial differential equations, by Craig Evans, publishedin 1998 by the Ameri can Mat hematical Society2. Chapter 9 of Real and Complex Analysis by Walter Rudin, published byMcGraw Hill3. The Fourier Integ ral and Certain of it Applicat i ons, by Norbert Weiner,published by