BASIC FOURIER TRANSFORMS

APPROVED*

J- /0^U^K-\l, ,i « iVljf <1 iajor Prof»slute 7^

fl. 1. fic iiaor Pa?ofess<

-.. C. (f/XrtnJLi _ fflr<ijjfcqg of tt£ Department of Mathematics

y /% . ^ ' ji

>an of the Graduate School

BASIC FOURBSR THA13F0RMS

THESIS

Prasantad to tha Oraduata Council of tha

North f e x a s Stat® U n l r a r « i t y i n P a r t i a l

Ful f i l lment of tha Raquirananta

Foap t h e Degree of

MASTER OF AlfS

%

Jane* Randolph Cumbia, B. A.

Denton, Texas

January, 1962

fmm OF CONTESTS

Chapter

I. THE LSBB3QUS INTEGRAL « 1

ix, FouRiBR f m m w o m s • • . , . 8

III. PALTOMO AID PARSEVAL FORMULAS 1$

BIBLIOGRAPHY . • . . . . 19

111

CHAPTER I

THE UEBBSQPB INTEGRAL

fha Fouriar ssrlas and tha Fouriar integral ara al-

tarnata axprassloas of tha Fouriar thaoraa* tha Mri«» balng

a Uniting aasa of tha intagral and rim Ttria*

Tha Fouriar latagr&l li a most baautiful mathematical rasult because of tha econwr of swans employed la obtaining a noat general rasult* One for® of tha intagral la used both to analyse and to iiBtbiiliiMMTo mention only sonorous fitoalloBfi tha propagation of aleatrls signal* along a telegraph wire, and the conductioa of heat by tha earth's cruet, aa subjects in thalr generality intractable without it, la to give but a faabla ldta of its importance.1

Tha purposa of this paper is to dsYslop soae of tha mora

basio Fourier transforms which are tha outgrowth of tha

Fourier theorem* Although oftan approaohad from tha stand*

point of tha series* this papar will approach tha theorem

froa tha standpoint of tha integral*

Zn ordar to astabllsh a reasonably coanplete and sy®~

netrieal theory of Fourier transforms it is neoeesary to

introduce tha oonospt of Lebesque integration* It Is not

within tho soopa of this work to prasant a systematic and

*G. A* Campbell and R. K* Foster, Fonrlar Integrals For Fraotleal Application (Haw York, l^HTTppT 3-4*

thorough d«T*lopMiit of the iattgrftli hontfef It

1« useful to pmtnt the basio definitions of ths Integral

•long with eertein lamas in order to provide sane spe-

eifie vsanipulative • properties whioh will be useful in tho

proof* of later theorens*

mtore L&bemum Integration eaa be defined it Is neees*

Hiy to introduoe the oonoept of "*easttrability" uhleh will

b# dofinod as follows!

Definition 1»1. If 8 io a aot of points on a finite

interval, (a»b), let H be the totaX length of a finite or

denunerable set of intervale enclosing all points of §*

fhf unter measure of S ie the greatest lower bound of aXX

possible values of M and is denoted as ®(S)« Imt S* be the

set of all points of (a»b) whieh do not belong to 8# Then

the quantity n(S) • b • a - n(S) will be oalled the inner

Measure of 8« whenever s(S) * »(3) the set Mill be said to

be Measurable and the measure of S will be given as »(&)•

Definition 1*2. Over an infinite interval a set S

will be oalled weasurable whenever the portion of S in

every finite interval (a*b) is Beasurable where a^-b« If

g(a,b) represents this portion then the aeasure of I will

be defined ast »{ft) « II* mfjMa^bj]whenever this Unit CL-3*- CO «— b - ^ + o o

exists**

2Korb.rt Jte f w U r jatggf^ Aod SL Its AppXleations Ilfew Xoi«# P* oi

Definition 1«3. A set whleh la »easurable with respeot

to an outer measure as given In definition 1*1 la said to be

Lebesfue measurable.

How let f{x) be a real function of a real variable de-

fined on 3 and let h and k be two real numbers* The subset

of S whleh consists of those points x for whloh h<f <x)<k

will be represented by s[k<f(x}^.kjt

Definition l»ii> A funotlon f(x) Is said to be s»esur-

able If for every pair of real numbers h and k (h<k), the

set S{h<f(x) < I) Is measurable•

The Lebesque Integral for real funetlons of one real

variable oan now be defined.

Definition l»5f. Let y*f (x) be a bounded measurable

real funotlon defined on (a,b)« Let L be any number less

than the greatest lower bound of f (x) and let ¥ be any real

number greater than the least upper bound of f (x). Let the

Interval (L»U) be divided Into n parts by the partition "J*

whleh Is given by* L « y0 , y, , jz, •••»yn, « U suoh that

7 (1 « 1# 2, 3#

The upper sta» of f(x) on (a«b) as to 3 Is defined to be

^ J*Xy;»]s{T t< f (x)< y, )1 f the lower sua of f (x) on (atb)

<up to J i» d.fln.4 u K,, ((iXj, )] "her.

n[S] represents the Lebesque measure of the set $• The upper

Lebeague Integral of f(x) on (a,b) Is the greatest lower

%enry P. Thlelaum. Theory of Funetlons of Real Var-iables (Mew York, 1953)# p. 167. ' ^

bound of all upper •una of f (x) on (a#b) fox* all partition*

3 of (L,U)» The lower Lobeague integral of f(x) on (a,b)

la the leaat uppar bound of all lower tuna of f(x) on (a,b)

for all partitlone 3 of (L,U). If the uppar Lebeaqua in-

tagral of f (x) on (a9b) la equal to the lowar Lebeaqua

Integral of f (x) on (a*b)# their aommon value la eallad tha

Labaaqua intagral of f (x) on (afb) and la written f (x)dx«k

The raatrlatlona that tha funotlona aa wall aa thalr

domains of dafinitlon are bounded will now ba removed#

Definition 1*6. (Unbounded funotlona over sata of

flnlta meaaure) Let f (x) be measurable on S and lat A and

B ba two raal numbara auah that 4<B» A naw funotion

f^g(x) on S ia now dafinad aa:

ra if f(x)<A fAn(x) « \ f(x) If A^f (x)^B

Lb if B<f(x)

Tha Labaaqua intagral of f(x) on S ia daflnad aai

II* C fAB(x)dx

provided thla limit axiata«

Lat S ba a aat of infinita meaeure) lat a and b ba raal

numbara auoh that a<b# fha lntaraaction of S and (a,b) will

ba danotad aa 3*(a,b)

Dafinitlon 1»7. (Unbounded funotlona over aata of

infinite maaaura) Lat f(x) be a aeaaurable, but not neoea-

sarlly a bounded funotion defined on S (not necessarily of

Ulbid., pp. 170-171.

finite measure). If S(x) h&« a Le basque Integral over

S*(a,b) for ©very pair of real numbers a and %, and If

lim *\ If (x)| dx ixlitii than f (x) Is said to be summable Js<««

over 8, If f (x) la aummable over 3 then 11m <\ f (x)dx axlata

and Is danotad bj j gf(x)dx which Is oallad tha Lebasque

lntagral of f (x)ȣ

Definition 1.8. if a function (boundad or unboondad)

daflnad on A (of flnlta or Infinite measure) has a Lebesque

Integral than f(x) la said to be summable over

Definition 1.9. If f (x) Is a complex valued funotlon*

that Is If f (x) » u(x) + lv(x) where u(x) and v(x) are

measurable real funetlona, than the Lebesque Integral of

f (x) la defined ast \ f(x)dx « \u(x)dx + tW(x)dx.^ J*J

Heneeforth when the word "Integration" Is uaed It will

be taken to mean "Lebesque Integration" and Integral" will

m a n "Lebesque Integral" .

If S if a set of points In two dimensions tha defl~

nltlon and properties of ^f(x,y)dxdy do not differ es» s

sentially from those of the simple Integral; In tha defi-

nition of measure "interval" la replaoed by "reotangle" and

"length" by "area". Suwtablllty receives & precisely SIt>ld,. pp. 17J-17U-. 6P»ld.. p. 17U-»

T* Beld. "An Introduction To The Theory Of The Lebesque Integral," leetures delivered at the University of Chieago during tha summer quarter of 1937 > p« 3k*

analogous definition as that fop the simple integral*3 Al-

though most properties of the si»ple integral are carried

over into two space* a few arise that are new and are

stated as follows!

X«l. If f (x,y) is summable in (x#y) then it is

Reiner# ©£• cit., p. !$•

9Ibid., p. 18. 1 0IMd.

11S. H. Hofeton, fh» Ttaaorr of ?uaotlons of A Baal ¥«»• l.bl. and th. Itworf oTfeSSfin 3«rl.» (Sow VorkT I957)7~ Vol. ITP.TI9. —

'J

suanable in x for almost all y, and if ? represents the

entire (x,y) plane, then

SpSf(*#y)dxdy « WSfCxtyJdx*^ » 2&> -oo 1

Leama 1,2. If f(x,y) is measurable, if |f (x,y)|^ F(x,y)

for all arguments and if any one of these three integrals "

exists ^P(x,y)dxdy $dy $ P(x,y)dx,

s<X> C0

^dx P(x#y)dy 1

-GO -00 then all the following Integrals exist and are Identical.10

"v 'f (x#y)dxdyt W S f(x#y)dx, -<*>

\dx j f (x#y)dy. r«o —oo

the following Mean Value Theorem is in the form as given

by Veierstrass and by Du Bois-!ey»ond for the ease where g(x)

has a tienann integral, k later proof was given by Hobson

in which the only restriction inposed on g(x) is that it be

suBBBable, whether bounded or unbounded*11

1»3. If f (x) is monotone and bounded in the linear

interval (afb) and if g(x) is suwmable (bounded or unbounded)

in that interval then*

(f(x)g(x)dx « f(a)(g(x}dx • f(b)Cg(x)dx. Ja- -fa. -c.

Although the theory and background of the Lebesque

integral presented here has not been compraheaaive, by any

means, the foregoing is sufficient for the study that

follows*

CHAPTER II

FOURIER TRANSFORMS

The Fourier transforms, or pairs of functions, to be

considered will to© shown to b© a consequence of Fourierf s

Integral Theorem for functions belonging to class "L". 4

function f (x) is said to belong to a class Lp in a ^ x ^ b if

it is measurable there and if Jf{x)( p dx oo, For sim-

plicity L1 is written as "L". Another useful notion is that

of "bounded variation", k function f(x) is said to be of

bounded variation if there exists a positive number P such

that for every partition (a - xc, x,, x2, z„ « b, ^ i

*jl-< < *>:) |f(xi-() ~ f(x^)]<p.1

Lot t(x) bdlong to I* (« 00| co )t t&Mtn poo

lin \f(x)sin(kx)dx * 0« v/-ao

Proofs Given a positive number e, X can be ohosen so

large that f Cx)|l dx e and ^ jf (x)| dx c e. Hence

^f(x)sin(kx)dx|^^Jf(x)| )sin(kx)||dx| ^ | f (x)) d x < e.

Similarly it can be shown that JJ If (x)sin(kx)dx|< e. Now the r*

Second Mean Value Theorem can be applied to \f(x)sin(kx}dx

for <-X^c<X) as followsi \ f(x)sln(kx)dx «* f («X) (sinClcxJdx

• f(X) sin(kx)dx « f(-x) > oo>(ko)j + f(X) j^ooa(kc) » eoi.fMlj # Then as k-^oo, f(x)sin{kx)dx-*-(>•

^-Thielman, 0£. cit., p# 163.

a

Now 11a *\f (x)Bin(kx)dx « iim Jf (x)sin(kx)dx • k-+<*> 4oa «-*«> -co

l*£<x>.ia(kx)d* • £(,).i»(ta>te - 0,

ox's li» [ f(x)sln(kx)dx * 0* K - * i »

fheorem 2«2»^ Let f{x) belong to L (» oof a ) # Then

a necessary and sufficient condition that

X V 500 ^ - 0 0

^ jf(t)ooa[^(x - t)]dt « a o ^ -OO

is that for any fixed dt *<», -

lln ^ jf(x + y) • f(x - y) » 2aj5^2USEl<iy « 0»

Theorem 2.3* (Fourier*s Integral Theorem) Lot f (t)

belong to L (- oo , ® ) . If f(t) If of bounded variation in

an interval inoluding the point x than

| [f(x • 0) * f(x • 0)J« Tf^If(t)ooa|jf(x - t)]dt.

Prooft Lot g(y) « f(x • y) • f(x - y) - f(x • 0) •

f(* • 0)» Than g(y) is of bounded variation over (0td) if

d la small enough, and g(y)-»-0 aa y-»G» Lot g(y) « g ,(y) -

gg(y) whore g,(y) and ga(y) ar® positive non-decreasing

bounded funotlona in (0,d) which tend to 0 as y-*@* If e > 0

there is an n> 0 such that if y £ n then g, (y)^e«

* ^g,ty)gfete4y

- g . l O ^ W d y • g ^ i i S M l d y •

• (g ,{y)£i5iiSLldy » g , (n)^ il&lxldv

Jn # pk *

+ (g ,(y>2MiMldy* Since |sin(v)/v|^l (v^O) 1 (g(y)liBlMiay|<e w i y

% , C, Titebnarsh, Introduction to the Theory of Fourier Integrals (Oxford, 1937) * p. 13. """

I

10

for all valuee of k« laving fixed a, g,(y)/y t# integrable -j

over (n,d) such, that liia (g ,(y)ffMIW,dy w 0, Therefore

U » ^ , ( y ) ^ W - 0. "stallarJ, it oan b e ah«n that

11m (g2 (yJ^y^^dy • 0. Hence lim Cg(y)8^'*^dy « ^ *0 ^ Jo

lla^d(f (x + y) + f (x - y) * f (x + ©) - f (x - 0)] **y^'dy « 0«

Or? 4r f*3/ 5 ' ) c o i (* • t)Jdt « iff (x • 0) • fix • 0)] • O —00

Corollary 2,3* If f(t) is continuous and of bounded

variation in an Interval (a,b) than * \ c°° r^°

f (x) « Tv)&j Njf (t)coa[/(x • tfldt#

the integral converging uniformly In an interval interior to (a#b).3

Hienoeforth all funetions will be aaauned to be eon*

tinuoua »o that Corollary 2»3 holds*

Theorem 2.k» If Fc(^} la the Fourier Cosine Transform

of f(x), that i. If y f ) - jf£<t)oo.<t/)dt, than f(») is

given bys f (x) • * c OOooa(/<x)d/.

Proofi If f(t) Is defined for positive values of x

only it may be defined for all values by the substitution

f(t) ® f(-t). Therefore ^f(t)eos(V(x « t)]dt • C<X> _ ~C0 r-oo

]f(-t)ooajV(x + t)Jdt « }f (t)©o®[)£(x -f t)Jdt«

Then tt^{f (t)ooe[j6(x - tfjdt » xr^^f (t)ooeQf(x - t)^dt •

• djrfjf (t)oo»[^(x # tf)dt « w^d/^f(t)2[cos(/x)coa(^t)]dt * I^-Cooi(/x)d/ f(t)eoe(t/)dt « c(j6)ao»(^x)d^.

3lbIda, p. 13,

11

t h e o r e m 2 » $ » I f F s ( / ) I s t h e F o w l e r S i n ® T r a n s f o r m

o f f i x ) , t h a t I s I f F s < / ) « ^ f ( t ) s i n ( / * t ) d t , t h e n f ( x > i ®

g i v e n b y t f ( x ) • ^ F s ( ^ ) s i n { / x ) d / ,

o

P r o o f s I f f ( t ) i s d e f i n e d f o r p o s i t I T S v a l u e s o f x

o n l y i t m a y b e d e f i n e d f o r a l l v a l u e s b y t h e s u b s t i t u t i o n

f ( t ) * • f { * t ) # T h e r e f o r e ^ f ( t ) o o s [ / f ( x * t Q d t »

pep -00 - c o

j f ( » t ) e o s [ V ( x % t ) J d t « - ^ f ( t ) e o s [ f f ( x * t j ^ d t *

, ° /"00 ^ 00 ° ^00 y-OO

T h e n ^ j f ( t ) e o s [ / ( x • t f ] d t * l f \ d ^ f ( t ) e o s | j l { x - t Q d t •

^ d ^ f < t ) o o s [ / f ( x • t f j d t • T f ^ d / ^ ( t ) 2 [ s i n ( ^ x ) s i n ( / t ) J d t •

\ j i ^ s i n ( / & t ) d ^ \ f | - 5 f ( t ) o o s ( t / ) d t « \ j ^ \ ? ^ ( ^ ) s i n ( / < x ) d / # % T h e o r e m 2 * 6 * I f F ( p ) i s t h e F o u r i e r T r a n s f o r m o f

f ( x ) , t h a t i s i f P ( ^ ) » t h e n f ( x ) i s g i v e n

b y t f ( x > • ^ F C ^ e - M * ^ , ° °

P r o o f i D e f i n i t i o n s o f e v e n a n d o d d f u n c t i o n s a n d o f

cmpL* T v l . b l . . y l . 1 4 . g i n & N x - t > ] * < - 0 ,

U o . [ > ( X - * ) 3 < V • 2 ^ o o » [ > f ( x - t ) 3 < / l 2 ^ « o « \ } ( ( x - t f J d / S 0 r ; - M ( x - * ) ^ + i C i i n r ^ , . t o ^ - - * » d > .

- M . / " ^ o o oo CO

f ( * ) » f \ ) f ( t ) « o s [ > ( x - t Q d ^ d t - ^ d t ^ f ( t ) o o s [ ^ ( x - t j d ^ l «

l i a ^ f ( t ) d t 5 oM e o s ^ ( x - t ) ] d / * l i f f i ^ r « A ^ ( t * x ) d / ( f < t ) d t

-CO -Leo ^ 'OT J-» - L C 0 0

- W - » — u p \ * \ v / i

00

v f ^ F ( / ) e - l / * d / ,

I f t h e s a m e t r a n s f o r m w h i c h t u r n e d f ( x ) i n t o P ( x ) i s

a p p l i e d t o P ( x ) i t s e l f , t h e f u n c t i o n f ( » x ) i s o b t a i n e d * A

b l . » . S n e d d o n * F o u r i e r T r a n s f o r m s { l e w Y o r k * 1 9 5 1 ) »

p p . 1 8 - 1 9 #

12

farther application of the transformation yields F(-x) and

a fourth f (x)« Therefore the transformation is of period

four.**

The Fourier transforms of olass 1? take on a form sim-

ilar to those of olass !>• For transforms of this elass the

concept of "limit in tha mean" (abbreviated "l.i.uu") is

useful* The equation l«i*m. fn(t) « f(t) means that fn(t)

and f (t) belong to L («• oo , cd ) and that*

llm(|f n(t) - f(t)|2dt » 0.6

Another valuable working tool is the Rless*Fiseher

Theorem,

Theorem 2*1 J Given a sequence {tn (xj\ of functions of

<- 00 f m ) for which CO

a, lim \ If m(x) - fn(x)|2dx « 0

there exists a function f (x) belonging to X? such that

The converse transition from *bB to "a" can be obtained 9

by applying the Mlnkowsl inequality}

^Jf(x) • g(x)| 2dx^fc^^Jf (x) l2dx^ • lg(x) 12dx|^

where f(x) and g(x) belong to L2« 2

The fundsiaental theorem for transforms of class h is Flancherel's Theorem which is stated as follows 1

^Wiener# oju cit.# p. l$.7«

%»avld Vernon Widder, The Laplace Transformation (Princeton* 19M>)» p« 80.

' 1 op* £lt» 1 p• 27• JIbld»f PP* 22| 27•

XI

If F(») boloag* to «n (• m § m ) tboa

tbo funetloa F(x) * ^ff(x)o~***dt town an tbo Fourier

trauefow of f M n t o t i «aA *•!««• to I? and*

f(*> « ia«B«4r(t)o***4t| f|F(*)|2ta » $|f(x)|2d* "iO ~°° 2* °°

gbooroK 2>9> If f (*) bolongo to tr (-co, oo) and

F(x) is lti Fourior tMuasfowi, tho following formula* holdt

# _ 1 |l r<° _ % to, f W = C t % \ w m f ^ m y. h u

Proof i Lot (f n(xj}*o a ooqoonoo of function* which U

©ostium©*** aoii of bQun&m& wiotion ©?#r * fl&ito Interval

U»*) «Hi *0*0 ol*ovho*o «a«h that lig^Jf n(*) • f (x)|2dx • 0.

Lot Wjx) tlion ^wJfn(x)|2dx •

^ > ) d u j f n (T>d» a ' ^ ( U T J V)]. ^ f > ) A o 2 i r f n ( u ) .

f|f„<u>i2au. -co

H»t la, U a $>„ <*>!*«* - rbo(*)|2«* « C|f0<u)l®a». w-*oo -w 1 *<* n 1 J oq' u

Siaila*ljr, °°| Fm|*) •?„<*)[2 4* « SllMtt) - M u ) | 2 <fcu

A 1 " . 4 ) 1" - f" < u )l a du- ™ °

application* of tho 8ol«s»Fisohor Thaoreni glvoi HfJjVmC*5 - Fn(*)|^d* « Of li»^Fn(x) - F<x)|

2dx « 0. mat

0 F (x) * 3* # J*•?£ # Fn (%)» n — • 00

9I»>M.. p. 69.

lOtlMtaMwh, j£. olt.. pp. 73-Tfc,

1It

Ir nco If r Is soma real value of xs ^F0 (x)dx « y^dx^f^ (uje^^du

88 C du.

rr if® jfclru i letting n—»-oo , ) P(x)dx • f{m)2—j~-«= da tine®

(•irtt - l)/iu belongs to I?, Henoe

F<*' = va I; g ( t t ) ! — z t 1 du-

In order to prove the second part of this theorem the

following leisna. Is needed* This same eonditlon will be

proven later for funotlons of elass L.

Imam 2»9>^ If F(x) 1b the Pourier transform of f (x)

and O(x) le the Pourier tranaform of g(x), thent pOO pCD

j P(x)0(x)dx o ^f (x)g<-x)dx.

Continuing the prooft Let g(x) « 1 for O^x^r and

g(x) « 0 for x<?0, x>r. Then by part "a" of this theorenu

o(*) *.fLa_r«i]ct - idt -

1 * r » u - - 1 «ir* • i n s B i rr^"> vjBisirTi 0 1 s —

, % . ft*lrx . «» Therefore G(-x) « a* # Substitution of thia result

into the equation given by Lenaa 2#9 yields t

fY I r® g"i*t ^ ^ r<*> «»ixt i ^f (x)dx « ^ F ( t ) ^ dt# or f<x) » c t ) - •

xlViener, £. «££•» p. 70•

CHAPTER III

FALTUHG AID PARSSVAL FORMULAS

An interesting function in the study of Fourier trans*

form® it the funotion t*g which is known as the "Fanning" ,

"resultant"» or "convolution"* The Faltung f#g of the

functions f and § belonging to L (« co9 cd ) is defined ass3'

f«f « y=jg(t)f (X - t)dt.

ffhcercai 3«1.^ If F(^) and 0(/) are the Fourier trans*

forms of f(t) and g(%) respectively, then the Fourier trans-

form of the product (F)(0) is the Faltung f«g, that isi

V^F(^)G(j*)e"*^*d^ » ^ar$g(t)f (x » t)dt« -co -co

Proof *^g(t)f(x • t)dt

^-CO - 0 0 J

Since the factor appears on both sides of the equation

it is obvious that division of the equation by y== doe# not

distroy the equality« Similar results for even and odd

functions follow fro® the simple manipulation of terms and

will be stated as additional theorems* -

theorem 3*2# If (t) and Oc(t) are the Fourier cosine

transform of f (/rf) and g(j) respectively, theni

^Fc(t)Oc(t)cos(xt)dt « | g(^)[f (|x - /I) * fix *

^Sneddon, 0£» »lt,., p. 23*

2Ibld., p. 2k.

15

16

c00

Proof i jqPc (t)Gc(t)eos (xt)dt •

c (t) cos (xt) dt (/) cos (/rft) djf »

(*){•«»&<* * fl4 cos [t Ox • jH)]]dt «

i g(/)d,4||-£pc (t) jcosjb{x * )] • ©osjt(lx • fi\)j]dt «

h&mu* • /D • r (x • Theorem 3»3* If Fc(t) 1# the Fourier cosine transform

of f (/rf) and Os (t) is the Pour lor sine transform of them

$ Fc(t)G5(t)sin<xt)dt « $CgO#)(/(l* • /I) - f (x • /j]d « roo

0

Proofi ^Fc(t)as(t)sin(xt)dt *

\ff$Fc(fc)sin(xt)dt *

§ - 1)3- «oi[t(x • /rf)Jdt «

I S0«</> [jt( I * - II) - f (x • |JQd^*

Theorem 3«k«3 If Ps(t) is the Fowler sin® transform

of t&) and Oc(t) is the Fourier eoslne transform of g(/),

thent

5?^(t )G c {t) sin (xt) dt « J $/(jf)[g(lx - 1) - g(x • j£)]d4*

The following formulas are known as the Parseval for-

mulas because of their similarity to Farseyal•s formula

|^£f(x)]ata - 4»f + %i<L* * b?)

1© the theory of Fourlsr series

Theorem 3*5* If Fc(t) and Qc(t) are the Fourier cosine

transforms of f (jrf) and g(/rf) respectively, thent

^(tX^CtJdt » $f^)g(^)¥. 3lbld«, p. 25*

^itehmarsh, ©g. oit., p. 51,

IT

Prooft $Pc(t)Gc<t)dt * ^Pc(t) dt|/| g (jf)oos(t/6)&/6 -

§g(j)qff%$Fc (t)cos(tjrf)d/ m

And in the case when f = g# the following result Is ob-

tained! j^[fc (t)]2dt a £[f (/*)]2d^.

Theore* 3*6* If P{t) and o(t) are the Fourier trans-

forms of f {*£) and g{/} respectively, theai r oo ^oo

i p(t)G(t)dt - \f{/*)gM)<M* -00 co

Proofi ^P(t)G(t)dt * ^$0(t)dt f «

Theorem 3«7»^ If P(t) Is the Pourler transform of f{^)#

them $"(*(*) I 2<** « \ |f(/)|2d^. -GO 00

Proofi Prom the definition of Integral! of complex

function® Is Chapter I, the following follow* s . f 0 0 - — (• oo

g(t) » v5fr\G(t)e~lxtdt » S(t)eixbdti -loo -00

*<*) " I gTtie^dt « f u j e ' ^ d t .

In the equation of Theorew 3,6 replace g{-/) toy g(/),

Howt (altTr^dt -*eo r» r°°- **.1 _

^f(/rf)<tf C K t Je^dt « ^jP(t)G(t)dt.

And if f « g, • rP(t)i(t)dt, ori -CO - o o

$°°|p(t)|2dt «C°°(f ( )| 2d^ "CO -oo

The preo©ding formulas are useful in evaluating certain

types of integrals, A specific example is given here*

la order to evaluate the Integral 50dat/f(x2+'b2) (x2**2)]

Consider the two integrals A » •*"b*oos(tx)dx and

%bid«, p» m *

18

/*oo

B » j »*to*fin(tx)dx. Integrating A by parts gives: A » [(-1/b)e""bx©os {t*Oo- (t/b) ••b*sin (tx) dx -

(-l/b)e""00 901 (oo ) 4* l/b - (tA)$«""b*»ln(tx)dx * l/b - (t/b)B» Similarly B * (t/b)A. Now solve A and B simultaneouslys

A » Cl/b) - (t2A2)Aj A (1 + t2/b2) • lAl A « b/(t2 + b2)f and B « t/(t2 • b2).,

Let f (x) « t # b x, then Pc (t) • \RF $flf (x)cos(tx)dx » e-bxoos (txjdx » A « \]f Lb/(t2 • b2)] . In the same

manner, If g(x) « #-**, then Gc(t) » \ff [V(t2 + a2)]. Sub-

stitution into the formula Fc(t)Gc(t)dt » °°f(/)g()d/ givesi (#[b/(t2 • *2)]}{l/|[a/<t2 • a2)Tdt «°$e*(a+b)xdx, ori 5eodt/f(tW) (t2+a2)] « -(a+bjx . °

£••»(**)« / -(a • b)]® « 1/Ca + bh

Therefore dt/Qt b2)(t2+a2fl • T/[2ab(a • b)] which is the sane result obtained by neans of the Caloulus of residues.

BIBLIOmPHX

Books

Campbell, George A., and Foster, Roland H*» Pourior Integral® for Practical Applications» New York, D, Van Noetrand TIdBp&nj,fn571 '"\9l\M*

Churchill, Ruel V*, Fourier Series and Boundry Value Prob-lems, lew York, MeGraw^IHr^ooSrCQmpany, 15E7 t5P.

Franklin, Fblllp, Fourier Methods. lew York, MoGraw-Hill Book Company, Inc., 194V*

Hobson* E. W., Tbe Theory of Functions of a Real Variable and the ffaeory"of FouHerTa Series''« ioiSmeiTT leiTfoFE, Sover ImUications,' Inc., 195f,

Lighthill, M« J., Fourier Analysis and Generalised Functions, New York, Cambridge University Press, 195*3 •

Paley, Hammond S. A. C., and Wiener, Horbert, Fourier Trans-form® in the Complex Domain, Hew York, The Amerloan mmrnmloir Society, 19^.

Sneddon, Ian H., Fourier Transforms, New York, MeGraw»Hlll Book Company, Inc*, 1951*

Thielman, Henry P», Theory of Function® of Real Variables, lew York, PrentTeeSallT Xm.t m f *

Titchmarsh, B. C., Introduction to the Theory of Fourier Integrals, Oxford, Sxford tfnTversiiy frees, 1^37#

Wldder* David Vernon, The Laplace Transform, Princeton, Princton C n l « r « W Pri.s,

Wiener* Norbert, The Fourier Integral and Certain of its AppllcatlonsTHBew 'f briC Sover TubHcatibns, Ino.TX933 •

Unpublished Material

Held, W, T», "An Introduction to the Theory of the X*ebesque Integral," unpublished lectures, Department of Math-ematics, University of Chicago, Chicago, Illinois, 1937*

19