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Chapter 2. Fourier Analysis

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Chapter 2. Fourier Analysis CHAPTER TREFETHEN Chapter Fourier Analysis The Fourier transform The semidiscrete Fourier transform Interp olation and sinc functions The discrete Fourier transform Vectors and multiple space dimensions Notes and references Untwisting all the chains that tie the hidden soul of harmony JOHN MILTON LAllegro CHAPTER TREFETHEN The last chapter dealt with time dep endence and this one is motivated by space dep endence Later chapters will combine the two Fourier analysis touches almost every asp ect of partial dierential equa tions and their numerical solution Sometimes Fourier ideas enter into the analysis of a numerical algorithm derived from other principlesesp ecially in the stability analysis of nitedierence formulas Sometimes they underlie the design of the algorithm itselfsp ectral metho ds And sometimes the situ ation is a mixture of b oth as with iterativeandmultigrid metho ds for elliptic equations For one reason or another Fourier analysis will app ear in all of the remaining chapters of this b o ok The impact of Fourier analysis is also felt in many elds b esides dier ential equations and their numerical solution such as quantum mechanics crystallography signal pro cessing statistics and information theory There are four varieties of Fourier transform dep ending on whether the spatial domain is unb ounded or bounded continuous or discrete Name Space variable Transform variable Fourier transform unb ounded continuous continuous unb ounded Fourier series bounded continuous discrete unb ounded semidiscrete Fourier transform unb ounded discrete continuous b ounded or z transform discrete Fourier transform b ounded discrete discrete bounded DFT The second and third varieties are mathematically equivalent This chapter will describ e the essentials of these op erations emphasizing the parallels be tween them In discrete metho ds for partial dierential equations one lo oks for a representation that will converge to a solution of the continuous problem as the mesh is rened Our denitions are chosen so that the same kind of convergence holds also for the transforms Rigorous Fourier analysis is a highly technical and highly develop ed area of mathematics which dep ends heavily on the theory of Leb esgue measure and integration We shall make use of L and spaces but for the most part this chapter avoids the technicalities In particular a number of statements made in this chapter hold not at every point of a domain but almost everywhere everywhere but onaset of measure zero THE FOURIER TRANSFORM TREFETHEN The Fourier transform If ux is a Leb esguemeasurable function of x R the L norm of u is the nonnegative or innite real number Z h i kuk juxj dx The symbol L Ltwo denotes the set of all functions for whichthisintegral is nite L fu kuk g and L are the sets of functions having nite L andL norms Similarly L dened by Z juxjdx kuk sup juxj kuk x Note that since the L norm is the norm used in most applications b ecause of its many desirable prop erties we have reserved the symbol kk without a subscript for it The convolution of two functions u v is the function u v dened by Z Z u v x u v x ux y v y dy uy v x y dy assuming these integrals exist One way to think of u v is as a weighted moving average of values uxwithweights dened by v x or vice versa u L the Fourier transform of u is the function u dened For any by Z i x u F u e uxdx R The quantity is known as the wavenumber the spatial analog of frequency this integral converges in the usual sense for all For many functions u L R but there are situations where this is not true and in these cases one must interpret the integral as a limit in a certain L norm sense of integrals R M as M The reader interested in such details should consult the M various b o oks listed in the references If u L then u exists for every and is continuous with resp ect to According to the RiemannLeb esgue Lemma it also satises ju jas THE FOURIER TRANSFORM TREFETHEN x Figure Space and wavenumb er domains for the Fourier trans form compare Figures and The following theorem summarizes some of the fundamental prop erties of Fourier transforms THE FOURIER TRANSFORM Theorem If u L then the Fourier transform Z i x u F u e uxdx R b elongs to L also and u can b e recovered from u bythe inverse Fourier transform Z i x ux F u x e u d x R The L norms of u and u are related by Parsevals equality p ku k kuk d If u L and v L or vice versa then u v L and u v satises d u v u v These four equations are of such fundamental imp ortance that they are worth commenting on individually although it is assumed the reader has al ready been exp osed to Fourier analysis As mentioned in the intro duction to this chapter some of these prop ertiesnamely equations and hold merely for almost every value of x or In fact even if f z is a continuous function in L its Fourier transform may fail to converge at certain p oints x To ensure pointwise convergence one needs additional assumptions such as that f is of b ounded variation dened b elow b efore Theorem and b elongs to L These assumptions also ensure that at any point x where f has a jump discontinuity its Fourier transform converges to the average value f x f x THE FOURIER TRANSFORM TREFETHEN First of all indicates that u is a measure of the correlation of i x ux with the function e The idea behind Fourier analysis is to interpret i x uxas a sup erp osition of mono chromatic waves e with various wave num b ers andu represents the complex amplitude more precisely amplitude density with resp ect to of the comp onentofu at wave number Conversely expresses the synthesis of u x as a sup erp osition of its i x comp onents e each multiplied by the appropriate factor u The factor is a nuisance that could have been put in various places in our formulas but is hard to eliminate entirely Equation Parsevals equality is a statement of energy conserva tion the L energy of any signal ux is equal to the sum of the energies of p By energy we mean its comp onent vibrations except for the factor the square of the L norm Finally the convolution equation is p erhaps the most subtle of d the four The left side u v represents the strength of the wave number comp onent that results when u is convolved with v in other words the degree to which u and v beat in and out of phase with each other at wave number when multiplied together in reverse order with a varying oset Such b eating is caused b y a quadratic interaction of the wave number comp onent in u with the same comp onentofv hence the righthand side u v All of the assertions of Theorem can be veried in the following ex ample which the reader should study carefully EXAMPLE Bsplines Supp ose u is the function for x ux otherwise p Figure Then by wehave kuk and gives Z i x sin e i x e dx u i This functionu is called a sinc function more on these in x From and the indisp ensable identity Z sin s ds s p which can b e derived by complex contour integration we calculate kuk which conrms From the denition it is readily veried that in this example jxj for x u ux otherwise worth memorizing THE FOURIER TRANSFORM TREFETHEN u u x d u u u u x d u u u u u u x Figure The rst three Bsplines of Example and their Fourier trans forms and x for x u u ux jxj x for jxj otherwise and by and the corresp onding Fourier transforms must b e sin sin d d u u u u u See Figure In general a convolution u of p copies of u has the Fourier transform p p sin ud Ffu u ug p Note that whenever u or any other function is convolved with the function u of p it b ecomes smo other since the convolution amounts to a lo cal moving average In particular u itself is piecewise continuous u u is continuous and has a piecewise continuous rst derivative u u u has a continuous derivative and a piecewise continuous second derivative and so on In general u is a piecewise p olynomial of degree p with a p con tinuous p nd derivative and a piecewise continuous p st derivative and is known as a Bspline See for example C de Bo or A Practical Guide to Splines Springer Thus convolution with u makes a function smo other while the eect on the Fourier transform is to multiply it bysin and therebymake it decay more rapidly This relationship is evident in Figure For applications to numerical metho ds for partial dierential equations there are two prop erties of the Fourier transform that are most imp ortant THE FOURIER TRANSFORM TREFETHEN One is equation the Fourier transform converts convolution into mul tiplication The second can be derived by integration by parts Z Z i x i x c u e u xdx i e uxdx i u x x the Fourier transform assuming ux is smo oth and decays at That is converts dierentiation into multiplication by i This result is rigorously valid for any absolutely continuous function u L whose derivative b elongs to L Note that dierentiation makes a function less smo oth so the fact that it makes the Fourier transform decay less rapidly ts the pattern mentioned ab ove for convolution Figure EXAMPLE The function for x ux for x otherwise illustrated in Figure has Fourier transform Z Z i x i x u e dx e dx i sin i i i i e e e e i i whichisi times the Fourier transform of the triangular hat function
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