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by: Mrs. Dedric Little
Mrs. Dedric Little
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This 12 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 507 at Oregon State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/224452/mth-507-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15
Fourier Transform Summer 2006 Mth 507 Bent E Petersen These notes are somewhat informal and contain no proofs Some of the missing proofs are very delicate and interesting but my goal was just to give you a bit of a feeling for the scope of the subject In spite of the amount of material here we only scratch the surface The Fourier transform or if you prefer harmonic analysis is a very large eld of study Fourier series were studied and applied as early as 1750 by Daniel Bernoulli The Fourier transform was introduced by Fourier in his 1811 paper on the propogation of heat by a limiting argument starting from Fourier series The same argument occurs in many engineering textbooks today He further elaborated the theory of the Fourier transform in his major work on heat in 1822 Since that time the Fourier transform has been applied in many areas 7 almost periodic functions operational calculus quantum eld theory partial dilTerential equations and so forth Since 1811 many people have extended the Fourier transform The culmination of much of this work is Schwartz s beautiful theory of Fourier transforms of temperate distributions developed during 194571950 This theory include most of the earlier extensions One exception is the Fourier transform of Carleman circa 193571944 whose work in this area anticipates the theory of hyperfunctions If f E E 1 Rn we de ne the Fourier transform 3 f or y of f by 33 2 H S A m H w 6 e ilt5 gtfx dx 1 Herex xlgtx2gtquot39 gtxn h l H gt611 and lt gtxgt 61x1 nxn The trailing hat is used when the expression for f is suf ciently complex that an incon veniently wide hat would be required if overset More generally if u E B1Rquot the space of complex Borel measures on Rquot we de ne 176 e44 gt dx 2 M th 507 Fourier Transform As an example consider the unit mass located at the origin It is usually called the Dirac delta or delta function and is denoted by 5 It is easy to see that 5 1 If u E BRquot then there is a unique nite positive Borel measure l u l and a Borel function n such that lhl l and du hdlu The measure ml is called the variation of u The number u H l u l Rn is called the total variation of u The total variation is a norm with respect to which BlRquot is a Banach space Moreover 2 1Rquot is a closed subspace and the total variation on this subspace is just the usual 21 norm The basic facts are Theorem 1 Riemann Lebesgue lemma The Fourier transform is an infective continuous indeed norm decreasing linear map of BlRquot into BOOKquot and of El Rn into COOKquot Here BOOKquot is the space of complexivalued bounded continuous functions on Rquot and C0 Rn is the space of continuous complexivalued functions vanishing at in nity The space BOOKquot is a Banach space relative to the supremum norm and C0 Rn is a closed subspace By the RiesziMarkolT theorem the space BlRquot is isometrically isomorphic to the dual space of C0 Rn The proof of boundedness and uniform continuity of the Fourier transform above is fairly simple For the boundedness l ldlulllull 3 where is the total variation of 11 Thus S For the continuity note A A lt nx gt Wlt n l 2 smfl dwxwo 4 as n a 0 through a sequence by the Lebesgue dominated convergence theorem Since the estimate does not depend on i we have actually proved uniform continu ity To prove j vanishes at in nity we rst show by integration by parts that the state ment is true iff is in C Rn Then we use the density of C Rn in El Rn and the estimate in the rst part Bent E Petersen Page 2 Summer 2006 M th 507 Fourier Transform As a simple example consider the characteristic function of the interval 76161 where a gt 0 A calculation shows the Fourier transform is 2 sina 5 g which clearly vanishes at in nity The space BlRquot is actually a Banach algebra with the product V de ned by hduv hltxygt duxdvy he COW 6 We note ERRquot is an ideal in BlRquot and mm fxy awry f e 21W 1 e BRquot lt7 Fubini s theorem implies if u v E BRquot then dv Vdr 8 rmV m 9 The second relation shows the Fourier transform is a homomorphism of Banach and algebras Let D J be differentiation with respect to the jth coordinate Then if f and D J f are in El Rn we have DJ 76 716 10 and iff and xJf are in ERRquot we have M 76 iDj 11 Property 8 is very important since it provides a clue how to extend the Fourier transform to larger spaces by duality Property 9 is the basis of various regular ization and convergence arguments Property 10 shows that the Fourier trans form diagonalizes differential operators This observation is the basis for using the Fourier transform to develop an operational calculus or more modestly fractional derivatives Properties 10 and 11 show that regularity properties respectively Bent E Petersen Page 3 Summer 2006 M th 507 Fourier Transform growth at in nity of f is re ected by the growth at in nity respectively regularity of An important example of a Fourier transform is the Gauss7Weierstrass kernel Wee ourquot e e x 2A mar2W W 12 From property 2 we obtain for f E E 1 Rn ourquot eilt5 gt e e 5 Zd Wxy fy dy 13 Since fWy aly l the convolution in 13 may be regarded as an average of f with weight function W As 8 a 0 the weight function in the integral peaks sharply at y x Therefore it is not too surprising that the convolution converges to f x as 8 a 0 The convergence is in the sense of E 1inorm and also pointwise at each Lebesgue point of f and so in particular almost everywhere If we apply the Lebesgue Dominated Convergence theorem on the left side we obtain Theorem 2 Fourier inversion formula If f E E 1 Rn anal j E El Rn then M 27W eilt5 gt d 14 for each Lebesgue pointx off That is f 2min where g is de ned by gx gx In view of 14 we de ne the inverse Fourier transform of g E E 1 Rn by 36 ourquot eilt5rgtgltxgt dx 15 This transform has essentially the same properties as the Fourier transform of course The function ng gt is constant on the parallel hyperplanes lt ax gt t and is periodic in the direction of i with radian frequency l i Thus we may regard 14 as a decomposition of f into plane waves where measures the amplitude and phase of each plane wave The support of a measurable function is the smallest closed set outside of which it vanishes almost everywhere 7 the existence of such a set follows from the regularity of Lebesgue measure In view of 14 the support of j is called the spectrum of f It is denoted spec f Intuitively we feel that if f has no high frequency components then f cannot vary too rapidly This intuition is con rmed by Bent E Petersen Page 4 Summer 2006 M th 507 Fourier Transform Theorem 3 Bernstein s Inequality If f E EWRquot arid SPe f 5ERquotll l M 16 then f is analytic and 2 le x 2 SM2 Hin 17 jl Bemstein s inequality gives a bound for the response time or resolution of a band7 limited linear device Note the statement refers to the Fourier transform of a bounded function which we have not discussed yet We will take care of that in a moment When radiation for example xray radiation is diffracted by a crystal the mea sured intensities of the diffracted spectra yield essentially the Fourier transform of the electron density f in the crystal Knowledge of f gives a complete descrip tion of the arrangement of atoms in the crystal The phase problem of diffraction crystallography is that only the amplitude can be observed 7 the phase is not available The relation of the inverse transforms of A that is f and of is therefore of considerable interest It turns out that it is more convenient to study 27lt The inverse Fourier transform of is called the Patterson transform Pf of f If f E 2 Rn then the RiemanniLebesgue lemma implies that the Patterson trans form is a uniformly continuous function vanishing at in nity By using equations 9 and 14 we see easily that the Patterson transform is given by Pfx fxyny dy 18 The problem of diffraction crystallography now is to recover as much information about f as possible from Pf The fact that f is real in the case of crystallography together with other symmetry information can be brought to bear It is worth noting that in the case of acoustical techniques for example in ultrasound investigations the phases may be available to some extent so the information is more detailed than in the optical case I don t know if this fact is useful since the longer wavelengths lead to lower resolution Berti E Petersen Page 5 Summer 2006 M th 507 Fourier Transform As we noted the Fourier inversion formula 6 represents f as a sum of plane waves Other representations are possible For example if we write the inversion formula in spherical coordinates a r where r gt 0 and a 6 Sr 1 and the Euclidean volume element is r 1 drda we obtain fltx f frx dr 19 where x 2717 quotrquot 1 s ei lt gtrw dw 20 A calculation shows Afr 42f 21 that is we have written f as a sum of eigenfunctions of the Laplacian Let x0 6 Rquot By the studying the Fourier transform not of f but of d f where d is a smooth function with support in a neighborhood of x0 and x0 1 one is led to Hormander s notion of the singular spectrum or the wave front set of f This set consists of points x where f fails to be smooth at x and where 6 in a sense speci es the high frequency components of the Fourier transform which account for the singularity of f at x For functions or distributions for example which are smooth along a submanifold the wave front set lies in the normal bundle of the submanifold Let s return to equation 8 If we replace v by f actually by f dx where f E El Rn then 8 says that the transpose 3 BRquot a EWRquot 22 of the Fourier transform 3 1Rquot C0lRiquot 23 is just the Fourier transform as already de ned on BlRquot This fact suggests con sidering the Fourier transform on a subspace 5 of El Rn and then extending it by duality Of course we want a very strong topology on 5 so that we will have many continuous linear functionals that is a large dual space For the space 5 Schwartz chose the set of functions d E C Rn such that max sup la lgm XER 1x239quotD ltxgt 0quot 24 Bent E Petersen Page 6 Summer 2006 M th 507 Fourier Transform for each integer m 2 0 Here 06 061062 ocn is an ordered ntuple of non negative integers with weight loci 061 06 and D D D2 where 117 J Bx If Cm is the best value of the constant Cm and we set 25 gt0 then 5 is a complete metric space with translation invariant metric 61 Moreover properties 10 and l l of the Fourier transform and the Fourier inversion formula 14 imply that the Fourier transform is a homeomorphism of 5 onto 5 We extend the Fourier transform to the dual space 5 by duality lt3f gtlt gtltfgt fey My 26 By standard arguments the Fourier transform is now a bijection of 5 and is a homeomorphism for the standard dual topologies Ifl S p S 00 we have a natural inclusion EPURquot Q 5 de ned by setting lt m gt fx x dx f6 21W 6 5 27 The Fourier transform is therefore now de ned on EPURquot for l S p S 00 In the case p l the new de nition agrees with the previous one The case p 2 is classical Theorem 4 Parseval Plancherel theorem 27ETquot23 is an isometric isomor phism ofcg2 Rn onto itself In the context of 5 the Fourier inversion formula becomes f 27c quot f for f e M 28 Here the re ection operator gx g7x is extended to 5 by duality The ele ments of 5 are the temperate or tempered distributions of Schwartz The space 5 contains much more than 2 functions Indeed it contains all locally integrable periodic functions all locally integrable homogeneous functions many of the nite parts of divergent integrals of Hadamard and much much more Bent E Petersen Page 7 Summer 2006 M th 507 Fourier Transform If f is a temperate distribution we de ne the support of f denoted supp f to be the smallest closed setA with the property that if d E 5 andA suppd 9 then lt f gt 0 This notion generalizes the notion of the support of a measurable func tion introduced above If f has compact support then its Fourier transform extends to a holomorphic func tion on Cquot The extension is given by z f ee iltzquotgtgt 29 where z 2122 2quot 9 is a smooth function with compact support and 9 l in a neighborhood of supp f The main result for Fourier transform of distributions with compact support is Theorem 5 Paley Wiener Theorem Let K be a compact convex subsets of Rquot with support function H Then a holomorphic function F on Cquot is the Fourier transform of a distribution with support in K ifanal only if there are constants C anal N such that Fz Clt1zNeHltquotgt 30 wherez in Here Hnsupltnxgt leK 31 It is sometimes useful to know the range of the Fourier transform On the spaces 5 5 and EZURquot the Fourier transform is bijective On distributions with compact support the paley7Wiener theorem gives a description of the range In the case of the Fourier transform of measures 3 BRquot a BOOKquot 32 the range is not closed and there appears to be no effective description of the range However there is an important partial result Let BRquot be the set of nonnegative nite Borel measures on Rquot Then BRquot is a closed convex cone in BlRquot and it is closed under the convolution product If f E CURquot then f is said to be of positive type iffor each N for each x1 xN E Rquot and for each 21 ZN E Cquot we have 2 N 2 fXJ fiegt212 2 0 33 lkl J Bent E Petersen Page 8 Summer 2006 M th 507 Fourier Transform It follows that f 7x f x and l f x l S f The functions of positive type form a closed cone FORquot in BOOKquot It is worth pointing out that on the real line any bounded positive even function which is convex on the half line is of positive type Note also that the Patterson transform Pfx fxyfy dy lt34 of an integrable function clearly is a function of positive type Indeed the Fourier transform of a positive measure is of postive type More precisely we have Theorem 6 Bochner s Theorem new FORquot 35 By the Jordan decomposition theorem each measure in BRquot is a linear combina tion of 4 measures in BJr Rn Thus FURquot spans the range of the Fourier transform and therefore provides an example of a closed convex cone which spans a nonclosed subspace One interesting application of the Fourier transform is to the operational calcu lus The problem is to assign a meaning to the symbol fA1 Aquot where f is a function or distribution in rt variables and where the A J are linear operators on a suitable space possibly noncommuting Weyl circa 1928 had the idea of de ning fltA1 Aquot ourquot eilt gtlt gt d 36 where lt 614 gt 1141 nAn The exponential is de ned in various standard ways For example if the operators Ak are bounded we can use the usual power series The integrand here is an operator and we interpret the integral as a vector7 integral In some contexts it is more convenient guided by the duality properties of the Fourier transform to move the Fourier transform over onto the exponential now regarded as an operatorvalued distribution and to de ne fltA1gtgtAnlt371lt671ltw4gtgtgtfgt 37 Bent E Petersen Page 9 Summer 2006 M th 507 Fourier Transform where we have used 27ETquot3g 371gv If the AJ are bounded operators on a Hilbert space then the inverse Fourier transform of the exponential is a distribution with compact supp01t and so the expression for fA1 Aquot makes sense for any smooth function f Let us look at a few very informal examples In various contexts these calculations can be xed up circumvented or justi ed Example 1 Let a E Rquot and letD D1D2 Dn where DJ as usual We J de ne the operator elt Dgt by means of the power series for the exponential Thus elt 1Dgt i lt agtD gtm 2 m a 1 EM 38 m0 where a a 3961quot and 06 061 an Then for a function u we have 1 elt Dgtux ZaDaubc za ux a 39 a where forgive me I have assumed u is the sum of its Taylor series Thus elt Dgt is just a translation operator Example 2 Consider now f7iD1 41Dquot conveniently written as f 41D By de nition and the previous example fiDux 2n quoteltquot Dgtnuxdn 2 quotnuxndn 40 If we formally compute the Fourier transform of f iiDu we obtain a nice ex pression fiDuA 2 e ilt5 gtnuxndndx lt41 ourquot eiltf quotgtn dn lt42 more 43 44 As a useful special case ifz is a complex number and A D D3l is the Laplacian then liAYuW 1l l2z 45 Bent E Petersen Page 10 Summer 2006 M th 507 Fourier Transform Example 3 In this example we deal with noncommuting operators For simplicity let rt 2 and consider the operators A1 and A2 de ned on functions on the line by A1ut tut 46 A2ut fitt 47 48 Let W fA1A2 and letD Then Wut my2 1 aemquotDur mg 49 To compute the exponential here we use the method of the integrating factor familiar from di erential equations We have in 6Dut e 9lt gtaD e9lt gtut 50 where 51 26 Thus eml TDuU e79lttgte6D e939gtut e79lttgte9lt gut 7 52 where we have use the result of example 1 Now i 16 9t679t11t17 53 Thus we obtain eiTt6D en39s2 eirt eoD 54 This is a nice example showing explicitly that the group property of the exponential does not hold for noncommuting exponents One can also obtain this formula from the CampbellHausdorff formulae We can obtain an explicit formula for W We have Wut MW1aequotlt3962gtutad1da 55 271 1ft626ut6 do 56 57 Bent E Petersen Page 11 Summer 2006 M th 507 Fourier Transform where we have inverted the Fourier transform relative to the rst variable Thus here f denotes the partial Fourier transform relative to the second variable Inserting the de nition of this partial Fourier transform we obtain Wut 27E 1e i ft62sut6dads 58 27E 1ei ylt rf sur drds 59 60 In particular if ft s ts then we obtain 7 is 7m imim Wut i 4 e se rur drds47r e ts e ur drds 61 emtuAs dstemsus ds 62 7DtuttDut lt63 64 We see we have the symmetric compromise between two equally reasonable de nitions of W One nal comment 7 in the case of periodic distributions the Fourier transform is a linear combination of point masses that is translates of the Dirac delta supported by the group of frequencies In this case the inverse Fourier transform becomes a series of eXponentials that is we obtain the Fourier series representation Thus we have come complete circle Copyright 1998 2006 Bent E Petersen Permission is granted to duplicate this document for nompro t educational purposes provided that no alterations are made and provided that this copyright notice is preserved on all copies Bent E Petersen 24 hour phone numbers Department of Mathematics of ce 541 7375163 Oregon State University Corvallis OR 973314605 fax 541 7370517 email petersenmathoregonstateedu httporegonstateeduNpeterseb Bent E Petersen Page 12 Summer 2006


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