BSTJ 40: 1. January 1961: Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I. (Slepian, D.; Pollak, H.O. 65-84)

7.5 MB·English

Prolate Spheroidal Wave Function:
Fourier Analysis and Uncertainty — I

Ry D. SLEPIAN and HO. FOLLAK

A complet i of teallnta functions ie decid eich poses the
comious propsty of Snény cba aver 0 gin ft eter a well ok
fer [once =). Prenton a the funations ave Gein aut rol pc
fans othe ropes a signa ae made

tie pointe out in tie paper ha the eiginfanesons of he Gite
Fourie: ravsfrm ane oeain prolate amhercil wave fanelinm There
cipunfunetioas propery extend pwesss properties that male them
lely site for the suly retain questions zegnrding the rusion
this beoween funetions al their Fourir (ransforme. Here we shell
nds the functions iv some dail aed peseent some apple ims to Ua
Fepreatation of bunllnie factions. The property thot we alle
Ineeteoneceed wth is Un ovthogonty of the fonesions over two dt
Tevont interes. "he ppes ay Late ed Poll eh fins dra
fon this materia, establishes ether properties wf che netiang se prox
Ses further espe af thei appheeton.

fre came definione won ial Hl essing, wo proceed to
stale niluat poof in Sexton TTT our rasa vate. Cerin appestions
(OC tives ers at then given in Seotion IV. The remaining eels of
‘he paper cee dear ots Thing rhe etl aleely sare.

In whet ola, we denote by £8 ease of all complex vatued ane
tian fC) defined om the rel line al istegrahle in absolae squire
Wefan then

0
Sh me muh sure erewtens aun, sasuany JO

fan raft [4 le ina energy of CE ee roe to Ft Lat
the oma of 0 a he iervot (AA), Tw ge manor, we
Uouote by 2 te sass of al eumplex uleedfunecions 8} dated for
SA2 18 A an iter in absolute are nears (=),

Fuuotions in 2. pass Fourier traseforms. Upper and lower care
versie of letter all evays denote « Louscr pain, We write, for exe

10 =) [rea ae, ®
Pw) ” ptieat (a)

Wo reer tof a6 tee, 13 angular frpusucy sx 2 a9 fraguoncy. Tae
functions Fo) ar aso integrable in sbsclute square. Ia this noterion
Pareles

[Cooma 2" rusts ao w

We denote by 0 th subolaw of 2, ennatting of thaw fess
40), Bhose Fourier seansiorma, Fla, varcah it [ay > 12. Here =
2x ia positive al number fed throughout thi pape. Rvery mewn
bor, 7) of a war be written as site Fonseetanform of fne
iow inergroble in abso sae

no =b [i rcoe an ,
‘From any funesion fiti in 2° we can obtain

funtion, BLO

=f [leona om

here #0) ie given by (i). We eal LV) the tanatnitod selon of
JO) Weonygant nsw yer ey lone Seton Sy 0
Pru hanna esi, Ti eleteen enginouring tera BVA)
results rum pessingf(¢} through ws eal low-pass Blter with engular
af frequen 8

‘We danote by 2 she subclass of feta, f(0, of 2 euch of whieh
vunishe for |1) > 2/2 Here P isa postive real number fixed through
cout tis wpe. Members of 3) ate called eli and 3 will bee
ferred the lace of feist Jumsone
om as fansrion () iv 8, nen obtaen a Funetion 1) contained
ingly the mle
(na, des 7a
bro! o
Uae
We eal Bi) the lime wien of 10), We eg 2 a6 cw operator
sheer eet on t. aneliva of ie to prone ie nited rare,
‘We aball sue tne station s(e) © 8 4 mex Uhst the fanetion $0)
belongs to he ease Tf Faetions,

“The statements weal Hy ate proved in Sections ¥ and VT
Given any > Mand any’ > U, wo em finds rounlably inénite cot
of rel fuvetions YH) (oo ad tof Peal postive nurers
Wo A> A> ey
vith the faowing properties
1. The ¥a(@) ave bate, oetbonarmal en the sea Tine al rom
plete in

~ fy ed

PRowwwe (8 ETE agama. oo

fi, Tor ail vies of 4 ro complex,

apiey = [nae

Word FBR. GD
‘Further proportion oti pare given in Sections Vand VT

“The notation sal above concerts the fees thw ele she and =
Is are funetions of the product 7, When i i menezary to male ths
epondence emp, we wee Ns = Ase}, le) = Had), = OLB,
‘where 20 = 97

‘ome vals of Aue} ate given sv Table T. This ta be noted thal for a
lived as ef echo 8 al ot bs neu rapidly wih seeing noe Ths
4h um ne avmra werusteet Joona, sastatne ING

‘Tyo T—Vanes ov Ayle) = Bylo) 10-66

es
[Penenence

exceeded (2/ne. (The senieange of the willbe discussed in seein
telecer paper) Beewawe of (0) ate (103, namely [Ys Let = 1 Ben?
hyn sl valu of 9; imple hal (0 ell ae pst of anergy
Sls the Interval (—1/2)0/2) whervan a valan of Ay new | implies
st y(2} will bo concentrated lage in (712,72), This bse
of the fs eon bo clurly ston in oes. 1 through i. Fig rough 4
show Wlsd. #.Cob), sie!) and ged) for anveral diferent, values of
ore ~ O3y or Qi/r}9 ~ UslLS shown on Hig. 1 Yo aa pr are pe
‘Healy aero in tho iserval (~ 1/2, /2). Core ~ 408 (2/n)e = 26,
as shown on Fig 4 yy i lngelyeonoeatzated in the interval (—1/2,
1/2). Fig. 6 compares Pe) for several diferent values of

41 Extapotation of Benatiitad Function

Ts pmetimes deste to estrapolate « hunted function known
nly om the intezeal (—7/2.2/2) to values outeda thie inarval. Sine
(hip /€ in am entire fnesion, tale extrapolecian san be done way
‘principle. One cou, Zor exsmple, cleuate seceeive derivatives of
Sabo point i {= 2''2,0/2) ard form » Taylor eres repevencation
‘which wand exaverge everywhere. In practic, however, uch a Taylor
erie would terssurly le truss tad the reakantproszation
{> Ad) moll em polynomial whic for ssbally lege values of
TE] sould give a vary poor sppeesimalion to This approxinsion i=
tng nf ce, bale

‘The functions provide an allerative apprnach, Since ¢ 2 8, we
enn write, fom orl ¢

10) = Fass, 1”

i] Ky)
* (th) i

al
fA

4
|
rant
SY nue mete svsnea meenteaL socnaL, senEaa 106

52 mn nrou aveerse sHemstexe soMmsAD, CaM TUBE

whee

a= Drona

: as)
Ear- [nora
and nye in 12) ea the ean squae ense
tin fo [io - Lasso] a =o
Mal ly (2) by #0 sega wow (10). Thee te
a= bf agen at asi

The evens in (12) om be dterined tg C14) from aoe of 10) te
ral (7/9, 2),
‘The above resul: suggests sppoesimating

Eaves os:

“The nppvexinatiot (15) in taf bandied,

1 for all ay

stl te
The ment suse ane i

Lu

ad ey (18) ean be mode ll desired by making 2 aeently
lange. In the sence of (16), tho extraplation renting good
he evar in ue of foto Fn {—B/2. 9) poem by

(insta = Bey

10)

pote = Sx

‘As tho Ny appeoeeh aero rapaly fr afisiently large mt
‘hat (17) inal for values of N fe which (16) il large. The fi of
fs inside the interval should not be Laken a a indialion of he Bit eke
shore,

4 Aymrdmatin tn a Puerta Baise? Bacon

Suppose now {CE © Wow he intareal (—7/24"/2) Sut
4a not nmewsearly a pee af Innate neon. From k above it