chapter 7 the fourier transform

231 Pages·2007·49.12 MB·English

CHAPTER 7
THE FOURIER TRANSFORM

Up to this point, the main heme ofthis 00K hasbeen the theory and application
of init sees expansions involsng various onhoncrival sets of funetcns We
‘bow turn tothe study of itegalanslorms dilfeent but related collection of
techmigues for unalyaing funcloas aad soviag diferential equations. We begin
With tie Founer trnsfon. vaich provides a way of expandig funetons on
the whole ral line R= {-sc.oct a (conliauous) superpositions of the Basic
‘oscillatory functions e®* (¢ © Rn much the same way thal Fourier series are
Used <0 expand funetions or a finite interval, Ta provide some motivation ll
1s pororm a few formal calculations

‘Sonpos that / iva function on R. For any {> U we cam expand f en
the interval [-iy/] in a Fourer series, and we wish to sce what happens to ths
‘xpansion as we let 1 — se. To this end, we Wnte the Founer expansion as
follows: For v= I-61

[venta

Let AS = nf and fe = AE = nas shen these formulas become

rr ita, at

cs

Prone 8 ay

Let us suppose that fc) vanishes rapidly a x — Loe thon 6) will not change
muti we extend the region of integration from [=i] to (es

eux [Sree

‘This last integral is a functicn only of, which we calla, and we now have

tee AE (llc

208
66 Other orthogonal hares 205

This noks very much like a Riemann sam, Ie now let foe. so that AF — 0,

sont tata = i seo ce as
foie [fie de, view tes= [foes mn

he niin ney nigh a oe
Fe corsair one tn ea
tone Te nan Fac een! ane 3
Finer he

erp re ila «com of aan omens We
cut lng tid ee an ow fer mee
“ibe dine ls eo as. aod, we toe at
Shlaesl gy silt ston ren tie no
i nr “

[vets

Moreover, 2 wil moan LR} the space of suare-ntgrable functions on
‘We ais inode the epace L! ~ 15 oF (absautely) itegeble Functions

™ wafr fipmnaess}

Utes, a8 with £2, the integral should 32 undorstood in the Lebesgue sens but
this technical point will not be of aay great coacern 10s) We remark that
iscota subset of /2, nore Ea subse a: ZL). The singuleres of» Funeton ix
Lh hat iy plies whore the alas of he function gene too: con he semevwhat
‘worse than those ofa funtion in, snoo aquerig w lage ramber makes i
large. o3 the ather hand, functions in Z2 nocd not docay es pil a infinity 35
these in. since squaring « small usher neki antler. For example, lel

f fecas

room {g” sae
om (y

Then f 1 mF bul aot io £2, wneeeas g is im Z? pt mo: ia L', (The easy
‘enfication i lll 10 te reader.) However. we have te flowing usefl facts:
(i) IEE L? and £4 bounded, then F'¢ E tadees,

fis = vies => [intarsy fester

(ai) 10 € 72 and f vanishes outside Gite in
follows from the Cauchy Schwarz ineuuiy

val aD}, then f = L!. Tals

fusiside= ft fonjaesih—al(f! sonra) 2
6 Chapter 7. The Fourier Transforin
7.1 Cowvolutions

efor stufying the Fourie transform, we need to introduce the convolution
>yedhet af tun Functions, device th wll be very usr both as a theortial
‘el and in apnlications, This iden mes seem abit mysterious tothe reader who
‘ac never stn it hefore; we sll ae afew c@ancnts on its meating fllowing
Thence 72, hut a fuller appreciation of ts signitcance esa best be achieved by
secing oi aries throughout the ecuirse of this ehepter

TT Fand are unetions on R, their comolution iste function f+ defined
hy

fray

provide vat the integral exits Vasious cousins car be imposed on f and g
to ensure that >a integral wll be absolutely convergent fr ally ¢ R for example:
(SM fed! and g is bourded say g <M, then

fi

(i IEF ie Bounded (say f]24f) and = 21, then

sapien dv ea | Vie vi dy = at fipidy <m

fisie-veonan eat fay ay

(ii) TE and ¢ are otk in £2, them bythe Caue-Schwateinegulit,

‘Jaro —siean

{iv} IF f1s plecewise coinuous ands is bounded and vanishes ouside afte
iojeral[e,2), then f+ gx) exits for all x, since the function > fr —99
is bounded on le 5] forary =

() Hoon be shown tha: if f and g are bath in L', then J+ glx} exists for
“almost every” xe, forall « exept for seme set having Labespt mesture
2910; moreover. J 7 ¢ L'. See Follené (25), $8.1, or Wheedea-Zyemund
6) 58,

‘This ist cam be extended. In what Follows we assune implicily thatthe funetions

ve mention satisfy appreprate conditions so thal all integrals in question are

absolutely convergent. The reader may supply specific bypotheses at wil Would
often be quite tedious tai all possible ones

‘We aow investigate the asic algera.c and aaalliceropertes of convalu-
tio

firecnatniars Two =n p<

Theorem 7. Convetion ays the same algebra fam as ordinary miplica-
or

Ai) fag — 5h
My feem ges
tus Feta oh

A 1+ LEO For amy constant

Fe ghee
Comvalations 207

Poa {i's obsious since negation is @ linear operation, For il. make
the change of varble = =~ »

fastain j pissy | fete aide 8-00
For i nd non: he oro incon
J eenee—vninay ff sorss

= [[ tows -2-vainaraee fF

Uses)

shy dy

hex Faigemiel

“Theorem 7.2. Suppave tat f ts dierentabie and the comoluons fg and f+
are wet-inec Then f+ is alferesuabie and (fs g\'= f= g Likewise, if §
is liforentighle then (f+ =f 0

‘Proof Jus ifferetiat under the itera sign:

feats J tx—oaonay— [ Po mieirids —

i},

Since f+ — ge.f the same argument works wth and g interchanged,

‘We emphasizs that in Theorem 7.2 one can throw the decivative in {f+ a
‘ono either factor. Thus f+ gis at east a6 smooth a5 either a, eveo when
the othe Factor has no smoo:hness properties

“Let us pause to makes few remarks that may sed some light athe meaning.
of convolations. Tn the fst place, Jet us think af the convolution iategeal as a
limi 0° Riemann sums,

ee Ee

the function Js(#} = Fx — 9} the Fametion F yrarslated along the sis hy
the ama 9, 80 the sm on The ght sea Tinear combination of yrncates
J with coeficents gly kts, Weecan theeefore thik of J» 2% 4 comtinwass
vperposiion of tmnslaies of f and since fe = a Fite also a continuous
Saperposiion of amines of

‘Second, convo tions may be ioterpete #5 “moving wright averages"
‘We recall tht the mverage valve of Function onthe inter [a,8] i dened
to be (b— ail ft fividy. More generally, the weighted average of f on [2,4]
‘vith egpect to & nonnegative wight Faction ie

Sfuwyrdy
ow dy

Suppose sow that gis nonnseatve and fgte}dy = 1. Irwe wte fs gtx) es
[ ivvetep}dy, we ace thas ngs 8 ns weighted average of (08 he wHle
208 Chapter 7. The Fourier Transform

line) with respect to the weight fonction wy} = gix~y). I g(x) ~Ofor|x| >a
then g(s=9)=0 for s=y)> 4,30 7 (8) isa weignted average of fon the
imtersal [e~ a, x +a]. To parteular, it

ate

(OP ean,"

ti
pened [sores

which i the (ordinary) average of fon the interval [x ~ a, x +a

‘One resect in which convolution docs rot resemble ordinary multiplication
is that whereas f= f fora. f, there is no function g such that f+ = f for
all f. (The Dirac “#-function” does the job, but it is neta genuine fenetion; we
‘hell discuss it in Chapter 8.) Havrever, we can easiy find sequences {+} Such
that f= fn converses tof asm + 9, The nition is provided by te remarks of
the preceding paranrazh If 2(x) vanishes (oe atleast is negligibly small) outside
anintereal t= a, then = g(x) will bea weighied average ofthe values of f
tn the interval (ra. +a), and ifa is very smal this should be approximately
fe

Tobe precise suppose # € Land for > Olet

aiar= te (2)

73)

That is, gi obtained from g by compressing the graph in the x-direction by @
factor of ¢ and ssmultaneously stretching it inthe y direction bya factor of Ie
(We are thinking ofthe case ¢ <1; ie > I the words compressing and stretching
should be interchanged. Ses Figure 7.1.) As ¢ ~ O the sranh of ge becomes @
Sharp spike at = 0, but the area under the graph remeins constant

fsimax= f2(2) 4(2) = f ainds.
Mor gcsenly the substiton x =ey yl
ff sinae~ [0° sordr. a

‘With this in mine, we can sate a precise theorem,

hore 73. Let § bean £8 fiction sac that (%,g(9)y = 8, anal et
J pO)dy ond B= fe aty}dy. (Noe tat ot B= and that a= B= 4
is din, Suppose thar} spleens contnucus on R and supose ether that fi
teounded or that g vanishes outside a ite interval so that f gx) 8 weledefned
forall x1 defined by (7.3), then

sl

3a Afixm) feral
FT Convolution: 208

‘rouRe 7. A function gs elt and ates ex) = df tate;
nl pled = 2g nea

1 pari, i comes a8 the
im fo ga) = for a

Moreower if contawons ai every point in the Dowadled intro [a3] the
‘cntergence i (753 's uniform on (2,0

Proc We have

femcad afiat)- Bila i= [fae feesair ay

+ [lim a te oases

0 we wish ta stow shar both isegralso» ie right san be made artery small
ds taking ¢soficendy smal, Tre argument is the same For hoth of them, s9 8c
‘consider only the second one. Given # > 9, we cap choase ¢ > 2 sll enous
Sothat for y) fl) <8 whom < v0, and wo brake vp the intra és
ht By Ty,

[flex 09 fe Jarra

28 [oi dene PP isinia
ee sings

ng e can oma a8 smal 38 We WSN Dy choosing sabe. To estimate

{ne ime eto, we tse “Re asumpion tha ether F bounds say

LAV¢ Af) or g vanishes outs de finite inerval tay gfx) =O for xj > RL TB
She ist case, by 24,

Plos-s

fst outs 2m [tay a0 fan
210 Chapter 7, The Fourier Transform

hica tends to zero along with ¢ I the sand ease. g(x) = for x > ei and
inpaniculr gin) for e> cif <eiR, so the integral fom 2 te stualy
‘auishes for ¢ salle

Finals i Fie continvous on [4,8 thea J is unifornly continuous thers
so the choie ofc -a che preceding argnmert cia be mace independent of for
1 lavil T follows easly thatthe convergence of x to F819 ure
ook 1

‘Those aro several variants of Theorem 7.3. which sav that fxge — Fn seme
sense or other as ©» O under suitable hspotheses. on f ands. Ws Stal conte:
furstves wilh slang a result for norm convergence of 1 fonctions

“Theorem 74. Suppase g © L! ts Bounded and sauiefes J e(vdy — 1 CF 21%
tho Foe} weld defined fr al, an if. dened eon 17.3), #2 converge
tof mann ase 9.

‘The proof of this result isnot cell diol, but i imwatves some anpraxi-
‘mation efguments that area bit beyond the level of the preset eiscusian. Soe
Folland [25], Thearera &.14, or Wheeden-Zjgraund (56), Theorem 9.6

‘Tao family a] in Theocems 7.3 und #4 i called an approsimate ideality,
since the operation of convatution with g tends to the ieatts operator as» ~
See Figuee 7.2,

FlouRE 7.2. 4 function f with an inate sngutarity and a jump diseot:
uit Het}, Go, {middle and F + Gpq (ght). where G is he Gaussian
fe
FL Convolutions 211

‘One oe funstionsg cat is ast oen used inthis conten isthe Gaussian

oye 6
Fay = hems

(etae[ewtaanbars an

Gis son sos hn teed the gn Tico 74 wehae += =

Gana alates versions (i have the property that all hes devivatives are
Tnoandet integrable functions. Teed itis easly estaNiched by insiction tha
GEG) — Pipe where Aisa polynomial of degree and it fellows thes
|@%ip)] 2 Cher, with similar estimates tinenling some powers of «) For
Ge. Hence we can apply Theorems 7.3 and Tuts If fie any) bounded and
piecewise continuors, then /~ Ge ie of las C, end it approniatee f when
"is small. These comvolations may be regarded at “amend ott” of “emootted
fut” veins of, What we have Sevloged here iss methos of spproximat.ng
‘general “onetions fy amcoth ones url technical toon many stutions, In
parscuar it yields a proof of the following fundamental result.

‘The Weierstnss Appmosimation ‘Theorom, [ff is catimvens fet om fa B]
(av ee beso. them i the unferm lime ofplrnomtals a a,b]. Tt
foram > O there isa polynamial P such thar

“sp, fos

Pines.

Tioene 7.3. A comtinsous function on [28] (lft anda continuous ox
tension of te R ight

Proxfe stone £ 19 be a continyous function om the whole real “ine that
sonishes outside the “ater! [a 1,4 fate Figure 23, By Theowem 7.3,
JG.» Fniformly on [ab where is given by 17.6) Ths, given 5 > Oc
is suficieny small we have
22 Chapter 7, The Rosarier Transforms

A x ranges ove [a Band » ranges eves [2 1,31 1) be fe ranges over
the bounded sel fed] whers c= lab Tye amd d= ih a1 Iie, and the
Taylor series S3(- 19° '/al for 7 converges eiiney on thi ae 1 allows
sll that we can repos «707 the above inceral ya suitable Taslor
polynomial without changing the sntegral by more shan 46, T> other words, i X
Is suiciendy lange,

sop fix} = Post <a

where va
Pix ae SUA op ay
But Px) isa polytomial af degree 2%, as one can see by expanding fs ~ 92"

by the Binomial theorem:

ne VS’. oem aalietg [ese 8

‘The Gaussian isnot the only commonly used approimeseidertity, Another
one is nen be ,

FUN ag oyap

Dich, at we shal ee arises inthe soloion of the Dirichlet oreblem fora half=
Dane. Teshares wih G the properties of being oven and having derivatives of all
‘der chat are bounded integrable fonetins. so st also provides smocth approx
mations to general bonnded functions. Anothes approximate idamtty with hese
Drepenits md ao extsa one thst maker it pastizulaey useful in some stations,
‘given by

fe

= TP eeepc,
Kyy=f
io

for yi 1,

1 possesses derisatves of al uiders, even aly = 2 because e
‘o infinite uelee a approsces from the Ie ur —1 from che eight) and it
‘anishes outside Ihe bounded set |p -e 1, Hence the eonvolisions f+ Ky are
selhdefined for any piewowise eatinuous /. bounded or aot and thy provide
Smooth sppvosinia oom vy all suc f. Sune oer apalewsons cf K are given in
Eercsee? and

EXERCISES
1. Which of the following faction are in L!? in

bel
1.2 The Fowrier transform 213

2. Lat Susi = fx? wnere f <9 < 2, Show that J's in neither £! nor L3, but
that {can be expressed asthe sum of an Z! fimeton andl 1? Funetion

3 Lat fist =f iter, fix) = 0 omherwiee,

Compute f- fand fF,
B Let in) <e° fie~s}avir (1.3)and let g(4) = 53s, Compute sng
nd check dreily that og.» 2ga6c 1 3, ‘Note hat 2 f fled]

4. Lat fos =e and gi) Compute f+ g. :Hint: Complete the
square in the exponent and use the fact taat f~" ax = yx.

5, For #> 2, let fe} = piney Me, Show that f= fon. (ine Bist
0 Erorise das a warmup}

6, For #> Optet fs) = x HT) for x > Q and ffxi = 0 foe x <4. Show
that + f= fev. (Ulin: The integral defining i, can be reduced to the
‘neal forthe beta function,

4. Show that for any 6 > O thece is a function on R wick the folowing
Dropertes (id is of lass CH) 0g of | forall x, (Hs b= 1
ween U< 2 1. (iv lx) =O when e< dor x > 14d. (Him: Define
fay fie)= lif 485 £1444, 1s) =O otherwise. Show that f= K
‘oes the jb PK is ac in {78} amd = = $5

Show that Zoramy [7 one any 3 > 0 these is a function x such that Gg
ig of elas C1 lg voatanes outside a finie interval and {8} Fg) ©
Prcesd by the folswing ster

Lat Fis} — fia) if fal 2 NF}
PSiEN is stein lars.

, Shew that ¢= £'+Kz does the job if is a in (7.8) ands suiciently
small,

otherwise. Shows that IF fl <

7.2. The Fourier transform

1 Fis am rope funtion on Rs Foret ante the fetion Pom R
deiuea by °

Fisre fer pon ate
esta sete tensa of 7. partway when heat!
by aime cumlnatedexpeson, We sal alse oteasonally weit

lpia! =F

foe the Pout sansfnem off. This involves an ungrammatical ue of the
sempols.x and but is sometimes te clearest wy af expressing things

Since ¢-™ has absolntevslne I. the integral converges absoluels for al
sad defines bounded fonetion of

Wes = fi fends. a8