BSTJ 40: 1. January 1961: Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II. (Landau, H.J.; Pollak, H.O.)

5.9 MB·English

Prolate Spheroidal Wave Functions,
Fourier Analysis and Uncertainty —II

By HJ LANDAU

“

©, POLLAK,

‘The theory dele i te preidng paper ie appiad ts a number of
ventions euler a banded sips. Tn patna, if 0
festeccergy signa is give, te posse proportions of sis enrpy 1 0
‘site ime iterated afte fegancy Band 28 fred, 08 el ne
‘gate which dhe bes job of somatonrous Hime ent frequen camera
tration.

It es common experez in tae cumin il Mat ene en
not simultaneously vonline a unetion Jf) and te Fouvier transform
Fi) too sevedy. The st foliar atemient ofthis phenomenon ie
the Heivenbere weertednte priate: Ive rasane fhe titwespread
bye) by

[a= arto ra

[nore

to he Frequency epee 8 af FW) by

then, Ine any choice of ty and en OF & J. Tang P and @ eexnot, for
say Fouser Usastorm pair, be oll stall Bxuality wil old 10)
[and hence Pia} are gzusian, and ty and ay are chosen as the meane
ff Lsi0) amd [a9 # tin this ease both zor). This result, hile
nun ibe orereN mecameaL sores, saNeaBy 1931

demunstenting that our experince with timeimiting and bandlinsiting
is indeed eet bo watematial ruth, dows not auceeed in proving
very good uersianding of what is ey happening. We should The
‘0 kar just host ease one ean ame to smnltanen nisng bot
time and requoney, a shal he price sUbat ane hw to pay We ec
aberper meseure of the contesteations of se) and Fé) than ted
Mlorded by the abuve vavianows of [7.0 Pando) * a mere
Ivbiehs if pomsbley wll depend oa the behavior of ft) ia given nite
Tins inter, al of (u) in a given faite fenesney ban

‘hn rosy attempt to mer thie need wns made hy L.A, MaeCll,
sao satal 1840 proved. the fallowing previonsy nmplehed form
of che uncertainty pri:

i
[uae
flisere
soa
LO Pte tate_
Litera
then

‘his heen does iacleedeanphasize the behavior of f(t) and Fo) in
ven Tile intevale. "Tho quantity a, represeting the proportion
OF the tal enemy of ft} whieh i he Cime-nterval (ey fyb Te
‘sperally etisiying a2 « meucwre of the sproud of #7; ww she ulher
Nand, a: hae no immediate physi interpretation, & turer dieulty
vith (1) iy hal there are no funetans Tor which equality eat be
Zehleved, although i practice ie estima i ile go

Amore uefa fore Tie uneetainty priseple von sepace che

stow mma ny the proportion af chery af a) in 9 Hensesey
and, ha is, by hiner to That af os. This is done inthe
proseat paper, We sal se tha i
nee

[ice tor

‘ele #a.3) il be foul eis iy, ue inealty #8 e abomp and
iimctons yesiry conti willbe gives, ‘The opriead fanebowe 20)
wil always be real iss i the above stetemen:. the Frey ison
Fonterad as were The ste sunt holds i th eequeney hand
irda ean youre at aero out thea sop Fete ate
in general, epee

‘Eee sho special cage of oor rol arian iC = 1, oo thae al
ores! he owntained inv 8, and Fis) = O for fo] > 4 The
lhstion "if a is given, wb is the in'mim S72" can aoe bo 2
Sliced fein, whet Be the masicurs 2” Lets rho
the folleying mations The sare norma of i he total may of

[swe

‘owetiovting a fro Fphoices a frets DY whi i$ aestrinad

wit) 12

‘Atondtnstng w6onstios fyinss a function Lf shoe Louse te
torn agro Ihe Foutir nemsform of or | | 2, and vuniehes
safe] > Hh
Pas = fis ay

ro azo that en alternative axpresin for if given by

1 fey anelt
p=! fp ts

ayo! petal
1 ane Ae ig ap a tn a

nite nol thn nine a egy mb tnd ya le
{econ hise te gcse she nee
aia <b [4

tall

Bay a

{in paticun, «function. is ales handled f — 2, then by
‘his result | DF 7 3224. This, now, is just the special case of che
nserainty principle tick we sare heen stoking: Uf 2 — 1, then
aevy.

Ta the sonzel, we shall take « longor lel at Us forme and ie
Fgnifeance; lt us, however, state che fll result for all valace of w
sau s*

‘Theerows There i fauesion ¢ yuh hab | = 1, BF) = eeu

BF | ~ ander the follows vendita, and oy idee the tele
ing comin

ano, hen

2WocasVe, mhmose sl

BMV ae S01, whence eae 6 E606 Ve

4a whee Den VE

"The bony of ue preaeat poper will caver the fllowing roquence of
lupice:Geeton HI will seep she peopoties of timelttd ond hand=
lietea fuseions, gad the geome interpreta W shew prapertng
wales we quire, Section TTT eualaize he pray” af Haque Uo,
lisnasion af the es" fneions, sad 3 number of partinen: geephe
tnd aria wenger TF id vals poste extension a Pe
‘huey, wad icles Je inveresing rl a if Fineliited tation
stand Tinted thnolion are ees, iia abeays poribe 2 find a
"alles" eign fs ak Df =e end Bf = he Pally, Sting
sve wppienions othe reveting Mowry in fila Unvony data wane
Taisicn a antenna thes.
Rota pranHOlvet, KATE FUNEMOSS —1 a0

Wr ore nenoerned, Sa te prow paper th ce ooletion af fun

fons feG whieh axe square infogeable ex (ye), Thos fon a
Fiert pte, denote ye in which the tamer prc) is do
Exedy

tne = [rom ah.

aud | fF = Gf 2s ws

"The coletinn of tintin ted functions oom Tino aubepaee
ob ae neil Fane fe invited, 0 eof bf Turtbernore,
is bomplce, whic mesas thet i we bave a aqaeaco ofan Us,
Joe Daal | fo fa | lh then there a a funesion {2 rach tha
Vea,

‘ihecily he ome satemeaca may lw rt: shoot bendlmitod func
tioaay sy Tor a ennplee linet aubspaon 92. "T latter ale
Tent fellows frum Uw earbee one thnengh the Parva svatin for
outier faoslinngs 17 F aad @ are the Fair aastorm uF and
swepecivey, the

[comm n= J [roa an

Wee stall ell tv Fanesions f and g erga if

(ta 0
Novice thst 2 ail {= Lf are ortingowal, singe each exe vanes
Size the ocer on sha aot; by she Puesoon raion, aed f — Bf
tem alo oeebogetal

"The ister probe pera is dee the angle bebe two Fe>
tione aad ys fille: Li the Sehner inna, we knew that

lute Silat
4 Retiad |S Ge) Ie
sve kaos chat

Beth a
sl-tel
‘We may eh define pe 4 heer the faneton

1

a gb
70m pun snvns tueusican soos, saseate 166

' wg) — eoe> Hetio
@ Tet

‘he existe reuse Mand v for Figg? cam be reached auly Hf and g

sre propostonal fan Pra sytity Holle in ‘he Seivare iuaguetey)

and 92 ol

Sup row that £6 Gl ant @ & aml cht neithor fanetion
sisi ideteally. What eae we way ali The an eee tem?
‘The wie can waa mig Th woe vn Sf = by. Ta nce

sar a a Tinga spaoes, hie won neean fat fh inline
Sandie, and ie He nown co be impnsstle.| TF, then, the ange
fant vaniah, ea it be axbccarky small? This i the hey spl
‘hich aball eecupy us for zune tne. Let us censide, fet ct all,
fixed funetion fw ud we aritrary 3 © 2D. We mow thot <j) eane
nt vanish; ie (9) howaded wos from aonoe AF thera is agora
Towor Insel Tor 8), he i neon for sume partial fanaa

2 DPT hie een pip, dn ade the
Fatosing
Tamme f: 16 & si6 give, then

int olga) >

ef is

fais assumed by 9 — AD fc any prative constant:
‘Prac It gia foto, them

Reigad = Lied — (he
ee ee eee
Fi
(ia | = PE Me |

SSS Se rTe neg geen 0 es ers Ce
so
Tella PE Recs
Wis 2 n> Pah
is wo nuts evi i fOr, i alls Thar
ifs Bb

furany g © 9th ity sles g aol Df ere propane THis
proves the fons.

We pooeced naw to the eae of arly f Wand g@ ©, Tek
sy, for onvenionne it

at te)

is etuallyansumed by spene Fant ons Han the pees amd ers
eget ate ee now ave he Calling

‘Whew de Thone execs least angle bern @ ord, TR angle
cuneate" ig ad ia assumed By Yo = wh end Dp 3, MOR
i he iasgestrigenwale of (2), ml the corresponding egertune-

‘as Rye pvc eg It,

nin aU Dil
ee ii
i)
if 6493 — inf ow LEE y
2 r “

tan he inferane om he Me (6) i ta nee Fhe i
Font The ight 's. Tee shoo in she pseding ope that any
ti mey he expanvted in a oves, conergert in-L? meza, of the
eagerness fy 2),

fo Eos.
72a, seer mpesiear eta seman IS

ny = Seto,

St fll from he prmperieg of 12 ta

Hopf Det

Te
we PH = coc (Boab),
i Ler
Sli en som in te poeaing paper Ua Ay <n HE B ,
ica at
max (21%
GE)
iewutined oy = O3or 2 130 tthe nvm pie val
=I Di
vet PH,

ranaeiy oo4 | hig, ar ually assumed it Yo, and g ~ fy. ‘the
thepre is prove

‘We have thn dows Uaat ae to eubapes aul of
nave is muti woeeptO Ha eaten, sti by lace insane ale
Detneen Tn, that, in fae a nzlile aetna a
Tiled fineliie causot oven he very low together. With th aid af
Ihe rst ae re ehall 906 tho amtoriinty privepie whieh we are
sexking ill allo:

Tn peeparacion for the roming Ucorems, 8 must consider oxo
nthe agpoct of the rymves al ®, Taw eloge doo sed cowether
came to Ala up all uf 2°? The en epee questions which eomecen
te are the followings 40 Lega ~ ty | dean Cauley gave
funetin =F, what can fe Sense Hse ak Tks al i)
fy there esi nein >

hie

4? arhngonal Wo both 8: and tha, to
‘very ietinn in i gm bi? "Hho anew bo Ulase guosrons ane The
Jpubjrtn ofthe subsequent the lem,

Trou 8:1 jh) ina aueay soquunce of Sovetions ot the form fy —

an erate RRS em ent Soe hat
MOAN TF SRS
et bowtie dy & Dand by © 9 for cue n how Une Timing fhnetin
Fits ofthe fom d+ by where! © Dud fe ©
‘Proafs Vow wl fy = ay hy 80 may alo Wrice

[= The = Bog) Whe bb)
Here Bly +e £2, while by — Db, LB. Th moe fallow for the
fact tha! Ue fs form's Caney sequence thot the functions 6, — Ty
a; Hor
= fal =

b= Dg = (be = Dig) 1 | Dhy = dy = Dy $y

sr Uk

1 = Da = he = Dowd So Fa

Lat now; since fy — Dh} fermi Caney sequence, 9 dove [Gael
For

ee ee en

and hy Ten

Pim — bed VEG Olle

to tt
Sines fel i now & Chushy styuenne, Maa kw fusetion = oh sh
thet

ean.
hos if] ml a boll covey eat, Benes a0 doe oi
outst buvetion 0 Sar wie,

jo bbe
We Ive tla lows Ue Cakig fini af sms of ain one

nl pivos as nothing aes, but ealy,oues gata, a um of function
in and b, We suty ehbrevito this by saving siuply shat ob 2 ie
tod

uma: There av fia Fantiane a Shih ave onto
soma tad FB
TA mm muy sveraot nacusie4s unady sANUAGY TES

Proofs The Suaetione
1 HT2=ns (ST 22041
0 eleowbere

‘ve instances of fumetions net in @ — 2, since the postion of fy in
Te] > 1/238 not w pie of y vaunted laation, Loma 2 panzita
toe cp write the bes: upprorimstion to j, frum ob + fs tho form
by Poy whore by © Bund dy © 0; thea

ahha

tune sive Fansions in 2 deh aes oathoyoaal boa 8
"Thm ae in fact, some arash "tsa te” Eometnns 2" — at —
ahan int} By ne de nes his, however, ef any Telly sonveuie

prewstati foe eueh Suaeloas,

We begin ly restate theorem aaswane ol in Beta T
‘Phorm 2 There ine anetion FES meh Hak [p= te] Bf! =a

nud [[B¢| = a sine the Glowing criti, and en's aaier the
fal niin

1a =0, hen tS <I

2un<a< ve, who ses)

RUE Sa <1, when eos et eos PB ee VI.
Ltan4 wae 02 6 Vn

Proofs La i the foray of fetes fx ith [f= Vand
DE oy satel, for vach cage of. delecmine

Bel

‘We sll alba show, i each ease, char ony alte nfo le thx the
sperm ea he erabzed hy 4% appropriate fonetion. Wartace ox
Tt the sprercum Steel? cau he realize will weny fom eago to ce.

(Caae 1. 0. Ite ~ 0, the funily § ear eoatain no Tanetion th
gL Loesff qth? = 1 wo mom have? © S, whenve fs azn
Ipfie and wonishos for [| <1 only ff = 0, This ion sontecirton

‘To abe that fontaine with ses ut B aeirily Pome
tole sot