Math 412-501Theory of Partial Differential EquationsLecture 3-8:Properties of Fourier transforms.Complex form of Fourier seriesA Fourier series on the interval [−L, L]:a0+X∞n=1ancosnπxL+X∞n=1bnsinnπxL.A Fourier series in the complex form:X∞n=−∞cnexpinπxL.For any y ∈ R,eiy= cos y + i sin y, e−iy= cos y − i sin y,cos y =12(eiy+ e−iy), sin y =12i(eiy− e−iy).Hence both forms of the Fourier series areequivalent.For any n ∈ Z, let φn(x) = einπx/L. Functions φnareorthogonal relative to the inner producthf , gi =ZL−Lf (x)g(x) dx.Indeed, if n 6= m, thenhφn, φmi =ZL−Leinπx/Leimπx/Ldx=ZL−Leinπx/Le−imπx/Ldx =ZL−Lei(n−m)πx /Ldx=Li(n − m)πei(n−m)πx /LL−L= 0.Also,hφn, φni =ZL−L|φn(x)|2dx =ZL−Ldx = 2L.Functions φnform a basis in the Hilbert spaceL2([−L, L]). Any square-integrable function f on[−L, L] is expanded into a seriesf (x) =X∞n=−∞cnφn(x) =X∞n=−∞cneinπx/Lthat converges in the mean. Coefficients areobtained as usual:cn=hf , φnihφn, φni=12LZL−Lf (x)e−inπx/Ldx.Fourier transformGiven a function h : R → C, the functionˆh(ω) = F[h](ω) =12πZ∞−∞h(x)e−iωxdx, ω ∈ Ris called the Fourier transform of h.Given a function H : R → C, the functionˇH(x) = F−1[H](x) =Z∞−∞H(ω)eiωxdω, x ∈ Ris called the inverse Fourier transform of H.Note that F−1[H](x) = 2π · F[H](−x).Discrepancy in the definitions“Mathematical” notation (used above):inner product: hf , gi =ZL−Lf (x)g(x) dx;Fourier coefficients:cn=hf , φnihφn, φni=12LZL−Lf (x)e−inπx/Ldx;Fourier transform:ˆf (ω) =12πZ∞−∞f (x)e−iωxdx;inverse Fourier transform:ˇF (x) =Z∞−∞F (ω)eiωxdω.Discrepancy in the definitions“Physical” notation (used by Haberman):inner (bra-ket) product: hf |gi =ZL−Lf (x)g(x) dx;Fourier coefficients:cn=hf |φnihφn|φni=12LZL−Lf (x)einπx/Ldx;Fourier transform:ˆf (ω) =12πZ∞−∞f (x)eiωxdx;inverse Fourier transform:ˇF (x) =Z∞−∞F (ω)e−iωxdω.Theorem Suppose h is an absolutely integrablefunction on (−∞, ∞) and let H = F[h] be itsFourier transform.(i) If h is smooth then h = F−1[H].(ii) If h is piecewise smooth then the inverseFourier transform F−1[H] is equal to h at points ofcontinuity. OtherwiseF−1[H](x) =h(x+) + h(x−)2.In particular, any smooth, absolutely integrablefunction h : R → C is represented as a Fourierintegralh(x) =Z∞−∞H(ω)eiωxdω.Proposition 1(i) F[af + bg] = aF[f ] + bF[g] for all a, b ∈ C.(ii) If g(x) = f (x + α) then ˆg(ω) = eiαωˆf (ω).(iii) If h(x) = eiβxf (x) thenˆh(ω) =ˆf (ω − β).Proof of (ii): ˆg(ω) =12πZRf (x + α)e−iωxdx=eiαω2πZRf (x + α)e−iω(x+α)dx=eiαω2πZRf (˜x)e−iω˜xd˜x = eiαωˆf (ω).Example. f (x) =(1, |x| ≤ a,0, |x| > a.ˆf (ω) =12πZ∞−∞f (x)e−iωxdx =12πZa−ae−iωxdx= −12π · iωe−iωxa−a=eiaω− e−iaω2π · iω=sin aωπω, ω 6= 0.ˆf (0) =12πZa−adx =aπ= limω→0sin aωπω.ThereforeZ∞−∞sin aωπωeiωxdω =1, |x| ≤ a,1/2, |x| = a,0, |x| > a.Proposition 2 Suppose thatR∞−∞|f (x)| dx < ∞.Then (i)ˆf is well defined and bounded;(ii)ˆf is continuous;(iii)ˆf (ω) → 0 as ω → ∞.|ˆf (ω)| =12πZRf (x)e−iωxdx≤12πZR|f (x)| dxStatement (iii) holds if f = χ[−a,a].Shift theorem =⇒ (iii) holds for any f = χ[a,b].Linearity =⇒ (iii) holds for piecewise constant functions.Finally, for any ε > 0 there exists a piecewiseconstant function fεsuch thatR∞−∞|f − fε| dx < ε.Theorem 1 Let f be a smo oth function such thatboth f and f′are absolutely integrable on R. Then(i)bf′(ω) = iω ·ˆf (ω);(ii)ˆf (ω) = α(ω)/ω, where limω→∞α(ω) = 0.Proof of (i):bf′(ω) =12πZRf′(x)e−iωxdx=12πf (x)e−iωx∞x=−∞−12πZRf (x)(e−iωx)′dx=iω2πZRf (x)e−iωxdx = iω ·ˆf (ω).f and f′are absolutely integrable =⇒ limx→∞f (x) = 0Corollary Let f be a smooth function such thatf , f′, f′′, . . . , f(k)are all absolutely integrable on R.Then (i)df(k)(ω) = (iω)kˆf (ω);(ii)ˆf (ω) = α(ω)/ωk, where limω→∞α(ω) = 0.Theorem 2 Let f be a function on R such thatRR(1 + |x|k)|f (x)| dx < ∞ for some integer k ≥ 1.Then (i)ˆf is k times differentiable;(ii)ˆf(k)(ω) = (−i)kF[xkf (x)](ω).ConvolutionSuppose f , g : R → C are bounded, absolutelyintegrable functions. The function(f ∗ g)(x) =ZRf (y)g(x − y) dyis called the convolution of f and g.Lemma f ∗ g = g ∗ f .Proof: Let z = x − y. Then(f ∗ g)(x) =Z∞−∞f (y)g(x − y) dy=Z∞−∞f (x − z)g(z) dz = (g ∗ f )(x).Convolution Theorem(i) F[f · g] = F[f ] ∗ F[g];(ii) F[f ∗ g] = 2π F[f ] · F[g].Proof of (ii): F[f ∗ g](ω) =12πZR(f ∗ g)(x)e−iωxdx=12πZRZRf (y)g(x − y)e−iωxdx dy (x = y + z)=12πZRZRf (y)g(z)e−iω(y +z)dz dy = 2πˆf (ω)ˆg(ω).Plancherel’s Theorem (a.k.a. Parseval’s Theorem)(i) If a function f is both absolutely integrable andsquare-integrable on R, then F[f ] is alsosquare-integrable. Moreover,ZR|f (x)|2dx = 2πZR|ˆf (ω)|2dω.(ii) If functions f , g are absolutely int egrable andsquare-integrable on R, thenZRf (x)g(x) dx = 2πZRˆf (ω)ˆg(ω) dω.That is, hf , gi = 2π hˆf ,