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MATH 311-504Topics in Applied MathematicsLecture 3-12:Fourier series (continued).Fourier seriesDefinition. Fourier series is a series of the forma0+X∞n=1ancos nx +X∞n=1bnsin nx.To each integrable function F : [−π, π] → R weassociate a Fourier series such thata0=12πZπ−πF (x) dxand for n ≥ 1,an=1πZπ−πF (x) cos nx dx,bn=1πZπ−πF (x) sin nx dx.Convergence in the meanTheorem Fourier series of a continuous functionon [−π, π] co nverges to this function with res pectto the distancedist(f , g ) = kf − gk =Zπ−π|f (x) − g(x)|2dx1/2.However such convergence in the mean does notnecessarily impl y pointwise convergence.Questionsf (x) ∼ a0+∞Xn=1ancos nx +∞Xn=1bnsin nx• When does a Fourier ser ies convergeeverywhere? When does it conver ge uniformly?• If a Fourier series does no t converg e everywhere,then what is the s et of points where it converges?• If a Fourier series is associated to a functio n,then how do converg ence properties depend on thefunction?• If a Fourier series is associated to a functio n,then how does the sum of the seri es relate to thefunction?Answersf (x) ∼ a0+∞Xn=1ancos nx +∞Xn=1bnsin nx• Complete answers are never easy (and hardlypossible) when dealing with the Fo ur ier ser ies!• A Fourier series converges ev erywhere providedthat an→ 0 and bn→ 0 fast enough ( howeverfast decay is not necessary).• The Fourier series of a continuous functionconverges to this function almost everywhere.• The Fourier series associated to a f uncti onconverges everywher e provided that the function ispiecewise smooth (condi tion may be relaxed).Jump discontinuityPiecewise continuous = finitely manyjump discontinuitiesPiecewise smooth function(both function and its derivativeare piecewise continuous)Continuous, but not piecewise smooth functionPointwise convergenceTheorem Suppose F : [−π, π] → R is a piecewisesmoo th function. Then the Four ier series of Fconverges everywher e.The sum at a point x (−π < x < π) is equal toF (x) if F is continuous at x. Otherwise the sum isequal toF (x−) + F (x+)2.The sum at the points π and −π is equal toF (π−) + F (−π+)2.Function and its Fourier series (L = π)Example. Fourier series of the function F (x) = x.x ∼ 2X∞n=1(−1)n+1sin nxn= 2sin x −12sin 2x +13sin 3x −14sin 4x + · · ·The series converges to the function F (x) f or any−π < x < π.For x = π/2 we obtain:π4= 1 −13+15−17+ · · ·Example. Fourier series of the function f (x) = x2.Proposition Fourier series of an odd functioncontains only sines, while Fourier series of an evenfunction contains only cosines and a constant term.Theorem Suppose that a functionf : [−π, π] → R is continuous, piecewis e smooth,and f (−π) = f (π).Then the Fouri er series of f′can be obtained viaterm-by-term differentiation of the Fourier seriesof f .Example. Fourier series of the function f (x) = x2.x2∼ a0+ a1cos x + a2cos 2x + a3cos 3x + · · ·Term-by -term differentiation yields−a1sin x − 2a2sin 2x − 3a3sin 3x − 4a4sin 4x − · · ·This should be the Fouri er series of f′(x) = 2x,which is2x ∼ 4sin x −12sin 2x +13sin 3x −14sin 4x + · · ·.Hence an= (−1)n4n2for n ≥ 1.It remains to find a0=12πZπ−πx2dx =π23.Example. Fourier series of the function f (x) = x2.x2∼π23+ 4X∞n=1(−1)ncos nxn2=π23+ 4− cos x +14cos 2x −19cos 3x +116cos 4x − · · ·The series converges to f (x) for any −π ≤ x ≤ π.For x = 0 we obtain:π212= 1 −122+132−142+ · · ·For x = π we obtain:π26= 1 +122+132+142+ · · ·Gibbs’ phenomenonππππLeft graph: Fourier series of F (x) = 2x.Right graph: 12th partial sum of the series.The maximal value of the nth partial sum for largen is about 17.9% higher than the maximal value ofthe series. This is the so-called Gibbs’


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