!#"#$%&'()%#*+-,. !//'0!12
34#!5&(34!#(+-,768:9!;!<#(#*=%&>
?@%ABC$(4*'D
'FEG+HI2#(
JK!/L(#+MN;O:PO!O#(#*=%&>
QSR TVUGQWTVX=U1Y[Z\X[]^QS_a`cb1Y[UGZ\d
`iXjYlkhQSm]
I (−1)
QS_e]fZRhg
n5oqpsrutvwpyxLzK{}|\~n€x@psyrutK‚„ƒ„…Vpy††!v†‡‚ƒ„ˆK~)†
‰‹Š!ŒC!Ž!4‘ n}’G“•”w–˜—?™2š\™›
œ
Kžf›1—€“•Ÿ2 ‡¡“¢”w›l ‡›£“•›Cœ•¤–¥’2š\’“š¦’2 G›
–¥§›C’
™wœ•¨\™›Cœ¢“¢–©›
—=¨¦ª I , š¦’–¥¤™¨\œ¢“•š¦’“1t&¨›
™w«©–¬“•­®¤š¦“¢œ•–¬¯@Ÿ2—°› ±–©’h†²–¥’³
¤›£“¢”2¨² 2—Cv´†‡¨\¤›W†²–¥’³µ¤ ›C“¢”w¨‡ hš\™w™w«©–©³
šB“¢–©¨\’— w›C™›C’2 ¶¨’±“•”w›®’w¨\’w{
—°–©’w§Ÿw«©š\œ¢–¥“€¡µ¨¦ª·“•”w–˜—‹¤-š¦“¢œ•–¬¯Wš¦’ W¨\’W“¢”w›=«©¨²³
šB“•–¥¨’W¨¦ªs–¥“•— ›
–¥§›C’²¸Bš¦«©Ÿw›
—
v
p·¤¨\’2§V“•”w›“¢”w›
¨\œ•›C¤-—¹žf›™wœ¢¨B¸›–˜—‹“¢”2š¦“
–˜— ’w¨\’—°–©’w§\Ÿ2«©š\œ
vµºl›
š¦«˜—°¨j—¢”w¨Bž»“¢”2š¦“¼–¬ª λ –˜—¼š=™wŸwœ•›-–©¤š\§\–©’2š¦œ•¡jI›C–©§\›
’²¸$š\«¥Ÿ2› ¨\ª I &“¢”w›
’
v
|λ| >
(−1)
(−1)
(−1)
1
π
!6 ½¾$¿·À·ÁfÂÃyÄyÅLÀ·Æ}¿
Ç "„0:/V$'„4qss$È
$"##El#!qC$„4µSq!
$:yÉsA/''$Ê
Ë2$ÍÌ11$#‹$#!CE[q%&#‹'Î'‹'0!$'![W%&'¹!:D!/"#!L½
É-jÏÌj#Ð I (−1) . >-"4//ÏE!(Lj'1Ñ#Î4#'$µ°'/'E±
ÉyA#/'Ê$
(A0!$±H:EÒÓ#Î4#'$ÈÔA"#[s0!!/AÕ®Ö
':$×#\Î4'!# È0'D!GHq/‡Ø1½>È/'!"0!Sµ"B˹IA#wØ?ÒH)!"&‹
0!/H4/Ù#ÚCE&S#$'®H)4²D&'!V 0!:D/"V-"B»°'/''
0!#$/•(A$˶ÛÜ8(AL½-6‡O:Ýwޕ(Aµ"BG/'
‹I‡Ø?ÒH)"#‹=Hq4²DA
yÏDA"4/f'0!:D/'"½
ß»?
"#/5
/'Ej'
Wµ=!2à
"-)34$Iµ%$#0!á I (−1)
?0!!4/ÊH/'=@'0!:D/"ÈV/A²±Ò$V!q
0!:4/Ï
/5Sâ°
GÛ¥Ý2Þ5ã\½‹ä @
!q
ÈV±€!-$V!:D0l„£Ö
B'@!:D!!/'"#$'!¶/0!!
'?¶Ïåf:5/·Ô"42$/'D!=H¶!
%&#l$!
E¶D/)u'æÛ¥Owޕ½µ>ÈjuçÛ¬Oޕ(KÏj4jHq»#$S
"'²/'/'Eµ4
j!qC$' #!/ˀ/'/)
"BÒË'‹·!
-2
1PA6wè&½V>/'!"0¶$'==
"#ÕS':j€!=S!
=$\$'²/y"q!
(K
0!#$/f
!H/é
# !qL½Ùêë €!//'wØ?‹$ ²
/'EW€
!ì$j\Î4ÍÖ
$'!G I (−1) $4'$-0!:D/"/'GËí\îï‡ð‡ñòG'0!:-4/Ï5ÖF/½ Ç ó
ôôwóõVöB÷ÍøwùCú-öB÷ûÏüFýÙþÿ CùëüC÷ ¥ö$ýëýFû )üëö¦÷'
û  fó  ó p ó  ó p  ó p v
!ù #"$&%('Bý&ö )'+*2ø %¢öBý£ù£,ý †‡–¥’2³\›C–©§\›C’²¸Bš¦«©Ÿw›&›
–¥§›C’²¸\›³F“¢¨œ
&—°–©’w§\Ÿ2«©š\œ¼¸Bš¦«©Ÿw›\4 ‡›C“¢›Cœ•¤–¬{
’2š\’“
w(— -\›CžL{}—¢¡¤¤›£“•œ¢–˜³¦‡t&¨²›C™w«©–¬“•­\v
.
I (−1)
!
$?
"#/'$Ù؋-Ø?//&‡DÈ'µ$'Ù4q¼4¼' λ '¼j"?'Ë04CE
'0!:D/"#Ó I (−1) (¹$ |λ| > 1 ½çêëW€!//'‡Ø?l$42 I (−1) W/ÏزEAS
:D!
H/'SÏÌ)½Ò±!wD(„Ø πØ?//„‡Ø $42 I (−1) 'l2l/'²C[/!
$50!4/'ʲH/='@l
l$42Ï'ËIÓ!#l"q4l¼0!#/'E
"\$"
I\ØÖF
EAS\$
1Ë2$ÍÌËØ?Ï$Ò'#/=0!D/'"½
ß»V"#1$j€!/'/‡Ø?0[$$'!Lá
â£6wã[É-®
ÔA"®!
$:j$
ÏÌÒØ?!
®/'S:$j®/'/·Ô"4/„ ω
-#@HEGHE ½
â¢è!ã[êë (λ, x) 0[ω]
4S=$#$
ÏÌ A ($ ؋Ø?Ï$ λ ∈
½j>È/
(q؋®"
l$lB2$ λ(A) $
evals(A)
S²Ó$ λ x?∈Òevecs(A)
0!D/'"j A ½
â°<:ã[É-¹$$q!=#â "S'5j$
)‡ã·# ÏÌ A 's#
H:E At â A∗ ã„#W$"#/-B! x = (x , x , . . . x ) :$
n
$#V!/"#GD! [x , x , . . . x ]t ' Cn. 1 2
1
2
n
+ȦÌ&(Ø V:A"=1'j€"!f(%&#j\$A? I (−1) ½
m $'&)(+*-,.*-/(10324032 É- sinc €"#$'!Ò?\Î4@-€!/'/‡Ø?á
%
sinc(x) =
(
sin(πx)
,
πx
€
!
1,
x = 0.
€!
x 6= 0
57698 ;
í :Ëñ <9=4ïò2ð@±¶°S/ÏEЀ!
µ"/5µH4
æ¶ð 698 í>@? 8 í< 6 ï 8
Ø?B»‡DA#"#$Ë#²Ì&$'!h 'D$ÏD!(sÎ4#'$»&Ö
Î'u':$0$/'Ò !:D!/"#!½éÉ-#uS$#&Ó؋
»\D!/!q
\ÌA$ÏD!/'E±H:E±3I±%A0ȱ
":®â°
ËBÛ AÞ¢½©ãG%&#[S&#
² Hqh"
$ "S
²//'Eç
!/'Dh'å)
:$/Ô"4!GØ?Ï$ Hq!"AÖ
EË/5²E!(q':$0$/' Ø?'±'#Î4Ï$j'CD/-!Ø?Ï$Ò'0!"/:0!(
µ'4
EVÏåf5/AÔ"42$!y4
5/AÏåf:5/AÔ"4!#„$42
4²D!®&\ÕØ?'¶'0!"/Ï$'½É-SVFØ Ò"S
²/yS&#
Ø?Bu)»uW#W€"#!Ñ\Î4ÐHqwD½S34!jS!®'#€!2Ö
$'!L(wjÛ©Pwޕ(&Û¥O2ޕ(GÛ¥Ý2Þ¢½34!·D!/"&$„l'0!$'!L(w$¼€!/'/‡Ø?0
°S/'ESy$
-q5/'/'EG'q!CB:²½
m $'&)(+*-,.*-/(1032C2 I (−1) $# n × n ÏÌË\Î4€!/'/‡Ø?á
%
Ø?
ηij = ei−j
É-"(
I (−1)
(
I (−1) = [ηij ]ni,j=1
Z k

(
#
1
ek = + s k
sk =
sinc(x)dx
2
0
²±H)1¦Ì##
¶Ó$#1€
I
(−1)
1
+ S,
=
2
Ø?
á
n
S
I (−1)
'‹=€!//'‡Ø?0VIØÖFCEASjÉsA/''$ÊÈË2$ÍÌSsS#
0
 s1

s
S=
 ½½½ 2


s
−s1
0
s½ 1
½½
s

−s2 ... −sn−1
−s1 ... −sn−2 

0½
½½½ 
½½ ... −sn−3

...
s
...
0
n−2
n−3
-É "0!!"&-$'-4q( n−1
\
€
W
1Ë2$ÍÌG#Î4H)‡D!½ Ç H
CD!
S
$42 S ¹[
²/4
IØÖë
EAS\$'És&/Ï$Ê-$
ÏÌ)½sêëLØ ?Ø : $VS4wÖ
'Ê[l#S
!± S, Ø lØ?
' S @/5l S. >È/[$
Ë2$ÍÌ [ 1 ] = 1 uu∗, Ø?Ï$ u = (1, 1, .n. . , 1), 1ÓIWËÍÌf½[É-
!S)!2
'!G2 I (−1) -I\EÓ$W!"/'EA'y'-H)4²D&'!½
&è ½ÂÅ&ÆÀ·Æ}„¿
wÄÀÂÁ I (−1)
ß»V‡DVD
'Î4@!#"#B!//ÏE@ !?®''!Ó"Ó$ 1000ã‹4
$0:D/" I (−1) /'È'Ë$#j!)Ó'0!:4/ÏK/½ I (−1) . É-=°\
$42®ç/'/ WEÚ/ÚÓ/'!æ
0!:®/'?/G€!/'/‡Ø?l'/'E
€! $j€!/'/‡Ø?#0µ/'S:BCE!(H#"#IEÒ
"/'½
$ C2#032 ñ < λ x ñ! 8 ñ 6#" ñ 8%$ 6& ï > A = [ω] + M, ω & ñ'î( 8 ò M
ð*):ñ,+.-0/[ñ & : 6 < 6 8 x = (x , x , . . . x ) 8 ò kxk = 1. 1 =4ñ 8
1
2
n
2
2
n
1 X 8 ò
x
Re(λ) =
Im(λ) = x∗ M x
2 j=1
j
2 & ï‡ï >43„ß Ï$Ó$10!ÏD!±#B!G؋V4²D!1€!
"ȇ؋D!(
Ax = λx
λ = x∗ [ω] x + x∗ M x.
P
2 1
²/¼#
x∗ [ω] x = ω nj=1 xj x∗ M x
$42
'j"
'Ë04CE
' M -IØÖ "È
Ï$L½ÙÉ-1"/Ï-€!//'‡Ø?½
êFN4
$'"/(¹'S€!/'/‡Ø?®€! W/¶$42Ë$#@
²/4CË=
'0!:D/"# λ I (−1) '1!0:'D!½ËÉ-S/G²Ð/'±H)W"
Ð
€"
¹/A²/Ê $#È0!D/'"¹ I (−1) ¼€!//'wØ?ás²/q
¹f²B
'0!:D/"#Ë I (−1) [H)\FØ 9¶ n/2 ½hÉshG$'(#$Ë4µ
'0!:D/"#¹µI 1 Ë2$ÍÌ 1 n/2 0 â}Ø?'$µ"/'/''FE
ã-%A kxk = 1, '-€!/'/‡Ø? $2 x∗ 1 x ≤ n/2.
n − 1.
É-"Ò
"±2=0!:D/"S I (−1) /'EA20Ñ'æÒ!) /'È'0!:
/5hÓ
" $ç‡Ø?'0æ4Òæ'0!:D/'"Ó (−1) /'¶ 0!'4
Eµ2Ì&½ÙÉ-=
"/'K$' !Ó
‡Ø?$I42'f$j "
/'EË0!'4
ES0!D/'" λ = bi I (−1) , $ |b| > 1/π.
I (−1)
m%$'&)(+*-,.*-/( C2C2 êë
()¶
t =¶'0!:D!!=
A
!
q!'0¶ñ 6#" ñ 8%v$ ï2=î 8 [vï 0: , v6 î1,·...,
vn]
V (z) = v0 + v1 z + ... + vn z n .
> €"! f (x) ?$Ò$SH)V " ñ 8 ñ & <Fï & €!WË2$ÍÌ
c
S# n (4' m = c (€?// 1 ≤ i, j ≤ n (Ø?
ij
ck =
M = [mij ]
i−j
1
2π
Z
π
f (x)e−ikx dx,
k = 0, 1, ..., n − 1.
Fê l‹€!/'/‡Ø?#0/'SË&(2؋ ‡D! 4 S 'y És&/Ï$ʼ$
ÏÌ10!£Ö
$®H:El$#-!S/¦Ìl€"#!
½„êF®Ø?42€!//'‡Ø?(-:0!/#
0!µ#\Ö
‹j"B:E Ù''4/ ¹i/x
/"##G¹$=J·H)0!"#=:$0!$/Ø?#B
4?Hq±"
±HEÒ$È"#$-Ò/'0µØ?'Ò0!$!½
−π
$ C22 i 6 ð
" ñ 8 ñ & F< ï & ï > 5 3
x
2 & ï‡ï >43„ß»jØ:
1
2π
Z
f (x) =
i
x
S$'
}Eqá
π
f (x)e
−irx
dx =
Z
r
sin(πx)
dx, r = 0, 1, ..., n − 1.
πx
È+ ‡Ø´Ø V/ $42
Z π
Z
â•è&½Ï6wã
i
cos(rx) − i sin(rx)
dx =
−π
2π
0
x
r
0
sin(πx)
dx.
πx
„3 V4È' r = 0 (q@$[/}4¶'Vâ•èA½'6wã- i R π dx (
2π −π x
Ø?B»1Ô"4/¹@Ê#½G>Ð' r 6= 0 (·$ cos(rx) V&f½Ó%&±
$S/\}
x
4Ò
1â•è&½Ï6wã '-‡Ø
−π
â•è&½©è!ã
i
2π
Z
π
−i
sin(rx)
dx,
x
Ø?BS¹W!
E®':$0$/
-$:$0!$W¼!:"!"½êFW°(
$1'0!Ò-\D!L½‹É-"(yâ•è&½©è!ã Hq
â•è&½¥<:ã
+È‡Ø "HC$'"#$
1
π
x=
π
y
r
−π
Z
π
sin(rx)
dx
x
»â•è&½¥<:ã¦(#$W0!\
Ø?B±
wD?$1/'SË#½
Z
0
r
0
sin(πy)
dy,
πy
I (−1)
És»C$"#EÐ$Ò\$
'4:W# 0!æ#!qC$® (¼Ï®"#ÕS
"#ENÑËÍÌ T ≡ −iS C$L½ T Ó0!$´S H:E 1/x ½ êë±
"ÈS'$@Ñ4®&D$0!®¼Hq'0Ó0!$hHE¶G/Ï֕D/"
€"\$L½É-°\$ /'#€"/qW$€!//'‡Ø?0j$#!
±(A
&K
Ø?B±-SA/'Ò/'!0µ$#V/'-yÉ-!
è&½Ï61 ÛÏ6‡9wޕ½
Y $'/.$ C2+2 > µ 6 ð $ ? & ñ 6 : " 698 & ®ñ 6#" ñ 8
î ?ñ=ï > I (−1) <9=4ñ 8 |µ| >
13
π
2 & ï‡ï >43ÙJK
Hqµ04 (−1) ()Ø?
/•()
Ø?'!"#„/!
(µ,qu)0!#$/'FE(:
"S$42I b '!#0:µ='Dib,
½„êëÙb€//‡Ø?s€!
$1':$ÏFEGê (−1) x = ibx $4
1
x
x + x∗ Sx = ib kxk2 .
2
∗
&% #Ò$±'0!: @ x∗ Sx ±"
/ÏEç0!
E(¹ÏW€!//'wØ?
$42 0 = x∗ 1 x = 1 Pn x 2 . É-€ 1 x = 0, €
! Ø?'B»Ï
j=1 j
2
2
€!//'‡Ø?=$4 (η,
'
V
/
»'0!j S 2 (0, x) VÐ0!4'
x)
[ 1 ] ½cÉ-ËË4 (b, x) Ó 04S T. +È‡Ø ' U (z) $V2'0!#)!/ÏEA!S5/KA$@Ø?Ï$@l0!:D!\$! u ('€!//'wØ?
$42h6hËÚÊ
ç U (z). 3$#¶°G4 (0, u) Ó 0!4'Ë
S/'?$4j®"#a¹$®&!
$= 'jÊ
(LØ?'BuS²
[ 12 ]
$42l$#Ó" -$Ó&\ÕS:$® U (z) lÊ
u½»É-"(ÙØ Ó Ø?'
€!W!S@q!/ÏE&#!/ Û (z). +È‡Ø /\ V (z) Hq¶
U
(z)
=
(z
−
1)
Û
(z),
)/'EA!S5/¹q!'0Ñ$uÑD\$! v #æØ?'B 4®Ö$6±SuÊ
½
É- V (z) ²±Hq1°$± V (z) = (z + 1)V̂ (z) ½¼É-"( U (z)V (z) =
½y+ȇػy$4·Ï z = eiθ (w (z−1)(z + 1) =
(z−1)(z ½‹
+%&1)
Û
(z)V̂ (z)
?W0!$!- T (4' €!//'wØ?‹$4
f (θ) = 1/θ
2i sin(θ)
2i
2π
É-"(
Z
π
−π
sin(θ)
Û (eiθ )V̂ (eiθ )dθ = hT u, vi = hλu, vi
θ
Z
2iλ π
=
sin(θ)Û (eiθ )V̂ (eiθ )dθ.
2π −π
Z
π
sin(θ)
− λ sin(θ) Û (eiθ )V̂ (eiθ )dθ = 0.
θ
π
Ç H
D!G4 V̂ (z) ²çHqGB!
$hHqÓ:EÚq!/ÏE&#!/Ù0!Ò
n − 1. I1$#VB#! V̂ (z) = Û (z) W!H#B
−π
Z
−π
sin(θ)
− b sin(θ) |Û (eiθ )|2 dθ = 0.
θ
I (−1)
&% # u yl0!D/'"(2$‹)/'EA!S5/ Û (z) s!Ê#½„ê넀//‡Ø?K$
$È:$0!$GHq‡D!È q!
'$ÏD!!)G:$
D/ (−π, 0). @!
‡D!(
$¹°! sin(θ)|Û (eiθ )|2 '„/)!
''D!¹!l'CD/ (0, π). êë€!//'‡Ø?
$42 1 −b < 0 €!·!S θ 'V‹':$CD/ (0, π). É-'y'#/f$ b > 1/π
Òθ/'$#V#A£½
êëـ!/'/‡Ø?y€
! É-#!
è&½¬8=$¼Ê1¦Ì&/"#S¼=q!
H/''0!AÖ
D/"j I (−1) .
g /./ C22 ï & î}îKò 6 :Ëñ 8 ð 6 ï 8 ð n <9=4ñ : < & 6
I (−1) 6 ð 8 ï 8 ð 698" ?&î# & 3
<#
½ lÆ Ù¿&Æ fÆAÆ5À  S
n
É-h
4/È$#!
V$'! Ó 4/'!0!"±$ É-!
è&½©èµÛÍ6‡9wޕ½ ÉsB ¬S!
µD/'Ú!/ÏEЀ!l²/ÏÖëD/'"çJKH)0!"¦Ö
:$0!$H/€"#!Ó.â −π ( π ã\(Ø?'B hÒ!CB:±!´ç
±
"# 2π (s
@Ïj²VHq/'¶ f (x) = 1/x »#(·$Ó$
Ë2$ÍÌu0!#$2$çH:E −if (x) ½»êF ˀ!/'/‡Ø?0Ò$#!
±(¼Ø‹Ó//¹"
$j#B! S C$Ò S $W'Ê=$#jS
!hâ€Ø?'B@' nã¦(
Ò"
T $Sn −iS .
n
n
& ñÓð 6 : $ îñ Y $'/.$ 24032 1 =4ñhñ 6#" ñ 8
î ?4ñBðï > −iS = T (t )n
n
n r−s r,s=1
+ =4ñ & ñ
)
Z π
â¢<#½Ï6wã
1 −ikθ
1
e
dθ, k = 0, 1, ..., n − 1.
t =
k
2π
θ
2 & ï‡ï >43ÙJK f (x) = 1/x ½=3„'
()#$V$4 f (x) ÈË0!$ T . ß»
Ø?/'/A
‡Ø 4ÙÏ λ Ù®0!:D/"‹ T µ"/Ï$#/'FEj0²2$y$4n®
r
Ô"4/4µè&(: f (x)
µ"C B40!n0ˋ/
3 $'¼ (−π, π) ½
− λr
//ÏE 1 − λ A¹‹B40!'0!S!
-$4!È' (−π, π) €!¹:E
B!'jxλ . É-r -Ø?'//f'#/'ES$4//f0!:D/" T V
S/½
+È$$r¹//Aq$€//‡Ø?'0'0!/ Î'(:Hqn²"-$#"B:E
''4/s4C=‹:E¶':$0$/·Ø?!
S':$0$jG
&#"j Gq!/ÏEÖ
!S5/) 1/θ Î4Ï$½
+ȇØc€!ȲBhD!! v = [v , v , ..., v ]t Cn (K
&5[µq!/'EAÖ
1 2
n
/
−π
V (z) = [1, z, z 2 , ..., z n−1 ]v =
êë u â¢<#½©è!ã
Ø?
1
v
z = eiθ
½
C
n
n
X
vj z j−1 .
(Ò$1CB±#j=1
-
&"#!
1
hu, vi =
2π
Z
π
U (z)V (z)dθ,
−π
Cn
$'
Î4
I (−1)
> /'®€ â¢<&½'6wã¦(Ø 1!H&B'Ò4
Z π
1
â¢<#½¥<:ã
1
U (z)V (z)dθ.
hTn u, vi =
2π −π θ
+È‡Ø /\ λ ≤ λ ≤ ... ≤ λ H)=='0!:D/'"K$"
Ï$5Ë$
ÏÌ
(Ø?' 1!
2q!'0Ún'0!:D! x , x , ..., x (È !
q!'0
T
1
2
n
'n0!q!/ÏE&#!/
Xi (z) = [1, z, ..., z n−1 ]xi , 1 ≤ i ≤ n.
34!aâ¢<#½©è!ã‹Ø‹V‡D
Z π
â¢<#½¬8ã
1
Xi (z)Xj (z)dθ = δij , 1 ≤ i, j ≤ n.
2π −π
34!7<#½©<µØ‹V/
S‡D
Z π
â¢<#½©P!ã
1
1
Xi (z)Xj (z)dθ = δij λi , 1 ≤ i, j ≤ n,
2π −π θ
Ø?
δ =$#
1
!BI1/'$#½lÉ-"(yÏ λ '1h'0!:D/"#®¹µ"/ÏÖ
ij
$'/''FEÒ(LÑØ W/5' 4 f (θ) − λ µr "C1B40!W
0!Ñ1
!S
):²½yÉ-¼q!:¹µ"C‹Hq 1/λ â°' λ 6= 0ãr! 0 â°' λ = 0ã\½Ù+$'-$
µ"/Ï$#/'EA0Gâ¢<#½¬8ã H:E λ (#Ò$r I0 ri = j = r (#EA/r
r
Z π
â¢<#½¥;:ã
1
λ |X (z)|2 dθ = λ .
>±H:EGBIA'0
â¢<#½©Ý!ã
r
2π
r
r
uâ°<#½©P!ã\(#؋V0!
−π
i=j=r
Z π
1
1
|Xr (z)|2 dθ = λr .
2π −π θ
+È‡Ø "H#$\µâ°<#½¥;:㋀!aâ¢<#½©Ý!ã‹SH#B'Ó$4
1
2π
Z
π
1
− λr |Xr (z)|2 dθ = 0.
θ
È+ ‡ØiÏ f (θ) − λ '-−π
!CB:È0@'
$1'0!/ r
Z
π
(−π, π)
(4²E@Ï??q!Ï$'D(
1
− λr |Xr (z)|2 dθ
θ
"/VH)¹q!Ï$'D½·É-'y−π
's'q!
H#/(²Ø?#Bl#‡D!„!"s
$'!µH)!"&
$1
0!ÒB4#0!1 f (θ) − λ .
+ȦÌAj"#)!
l$ m ≥ 2.r É-#µ\Ì&
! (1 − θλ )/θ B40!='0!
!/ÏE θ θ (Ø?
θ = 0 θ = λ . ßÐrhØ?'//=
‡Øa$42
$'"#1 #!Úâ€24 m ≥ 21ã-/$ÓË2!:$r\$'!L½+È[$Ï
"#ÕS[Ò4/'®$S²
SØ?# λ ≥ 0, #W!Ê
'0!:D/'"1
-)-€! ia ( a 6= 0 ( a ∈ R (r#SE®A"ÙW4½„É-\€!(
S
' n−a â°
"S a > 0 ãÙ¹'0!:D/"# T Ll"/'/'ÏFEl0²2$Ù4
n
I (−1)
!VÔ"4/¹$Ñè&(y$ a µÚ'0!:D/"#?l"/'/'ÏFE0²2$[$4Ð!
Ô"4/Lè&½
%&
"S14 λ ≥ 0 (4#@\Î4
r
â¢<#½¥O:ã
ßÐV
?$42-$j€"\$'!
â¢<#½BA:ã
1
(f (θ) − λr ) .
2π
g(θ) =
h(θ)
h(θ) = g(θ) sin
\Î4H:E
θ − θ1
2
sin
θ − θ2
2
A??B40!1'0!Ò (−π, π) ½
ÉsË[Ø?:EÒ$'È'-$"#(q$'jÎ4
$4È' θ ( i = 1, 2 (
i
//·' (−π, π) (q$ θ−θ =/ (−π, π) ½=É-"θ( sin( θ−θ
#Î4h!
)
2
2
æB40!±'0!æ!/'EÐØ? θ−θ = 0 ×?¢½¥½æØ? θ = θ (‹€
(−π, π) ½±ä "#(
B0!µ'0!Ð!#/'Eu2V$Gq!'½@ !
Ô"i /'E(
i = 1,2ÐB0!g(θ)
0»!#/'EÑ1
Ëq!:$V/'½Éy@‡Øì4 h A
h(θ)
¹B4#0!
0!¹$ÈF؋lq!'('¹
"#ÕS‹$l#‡D!$4¼
0!S
#&‹‹B0!ÈV
Ë/'/0!Hq!AAS θ (:€! i = 1, 2 ½Ù+$'
h(θ)
$42-€!²B i (Ø 1²±B#&!
V '0!H)!
A&Ói -$W/"# θ (#€
j
½
j 6=+ȇi Ø´Ø?'
h
i
i
i
h(θ) =
Ø?
1
h1 (θ)h2 (θ).
2π
sin(θ/2) #
θ − θ2
) .
h1 (θ) =
h2 (θ) = (1 − θλr ) sin(
θ
2
²/'/Ù$4 θ = 0 θ = λ . 3„
²(y/\lBBI»G'0!Ð h H)!"&
r
½-ä "&²( h 1AÈB420[
0!@
θ (q#!?& h ½‹É-A"#( h &
θB4
1 0j01Ò
½Ùêë?˹$WBBI1 $j
0!Ó h2 H)"# θ ½¹%&' h
θ
1
A-B40!1
0!Ò θ (#Ø j#±!/ÏE®BBIGj'0!Ò2 h H)!"&1
½[ä "#²()Ø? θ < θ (f؋®2 4²D! 1 − θλ > 0 sin((θ − θ )/2)2 < 0 ½
θÉ-2"ÈØ?
( 2 'È#0:'D½ Ç r@l$È4L(4Ø?2 θ > θ (
θ( <θ 2 h(θ)
2'
æ
½
É
A
#
"
(
?
Ø
(
1−
0!θλ
$ÏD!r½Ë<É-0
€!
sin((θ
( h A−Vθ2 )/2)
[B4>#0!0
0!Ð θ (#θÐ>θ2 h(θ)
&
h
2
?B40!1'0!Ò (−π, π) ½
+È‡Ø "q!¼$ λ = λ K#µ"#/'$'/'ÏFE?0!$·4V!KÔ"4/
è&½34! i = r, r +1 r1 ≤ jr+1
(؋?4²D!$ R π g(θ)X (z)X (z)dθ =
≤n
−π
i
j
0
Z
(Hq"
I (−1)
Z π
Z
1
λr π
f (θ)Xi (z)Xj (z)dθ −
Xi (z)Xj (z)dθ
g(θ)Xi (z)Xj (z)dθ =
2π −π
2π −π
−π
= λi δij − λr δij
= (λi − λr )δij .
π
+È‡Ø Ï
(:$Ë$HqwD'0!/4¹Ê½„êë¹/'[Ê
lÏ
H)²"# i λ= r = λ ½ÙÉ-#\€!(
r+1
c0
r
Z
Ø?
i = r+1
π
g(θ)p(z)Xj (z)dθ = 0, 1 ≤ j ≤ n,
−π
p(z) = c0 Xr (z) + c1 Xr+1 (z)
c1
?!CB:(Ø?BË-‹HqÊ#(&SØ?#BS$
}E
c0 Xr (1) + c1 Xr+1 (1) = 0.
É--S²-$4 p(z) '?SÊS)!/ÏEA!S5/¢½
+ȇØc
Q(z) = p(z)(z − eiθ ) (4Ø?Bq!/'EA/f0!
[/
$4±-Ô"4/K$ n − 1 (4z−±1!H#$G4
2
â¢<#½Ï6‡9:ã
Z
É-"(
Z
êëy؋V/
π
g(θ)p(z)Q(z)dθ = 0.
−π
π
g(θ)|p(z)|
−π
2
z − e−iθ2
z−1
dθ = 0.
P
1 c X (z) 2
l r+l
g1 (θ) = g(θ) l=0
,
z−1
$±VHqwD1Ô"4!ÒHq
â¢<#½Ï6!6wã
+È‡Ø Ï
Z
($
z = eiθ
+È(#4
π
g1 (θ)(z − 1)(z − e−iθ2 )dθ = 0.
−π
θ2
θ
θ − θ2
(z − 1)(z − e−iθ2 ) = 4e−i 2 sin( ) sin(
).
2
2
eiθ eiφ = ei(
θ+φ
)
2
(ei(
θ−φ
)
2
− e−i(
θ−φ
)
2
= 2ei(
θ+φ
)
2
sin(θ − φ).
(
É-
€!
(sâ°<#½'66wã H)!S
â¢<#½Ï6wè!ã
4
É-"(
â¢<#½Ï6‡<:ã
Z
π
g1 (θ) e
−π
Z
É--'#/‹4
Z
Ø?
−i
θ2
2
I (−1)
.
θ
θ − θ2
sin( ) sin(
) dθ = 0.
2
2
π
θ
θ − θ2
g1 (θ) sin( ) sin(
)dθ = 0.
2
2
−π
π
θ − θ2
θ
) dθ = 0.
α(z)g(θ) sin( ) sin(
2
2
−π
É-
€!
(
Z
p(z) 2
.
α(z) = z − 1
π
α(z)h(θ)dθ = 0.
%&# h &--B0!1'0!Ó (−π, π) α(z) ≥ 0 (#ÒÏ-µ"C-H)
$42 p(z)/(z
½ÙÉ-
€!
( p ≡ 0 ½ÙÉ-'--®!:$$\$GØ?B
−
1)
≡
0
‡D?$4-1µ"/Ï$#/'FEW„:EÓ0!D/'"j 1.
É-?Ô"4/' S = T I (−1) = [ 1 ] + S ‡Ø $4 I (−1) Ù2¼!C¹
$#Iµ!-"q4‡Ø ²E®€! 1ÏÌ[2Ø?'WS/''0!:D/"#(A¼CB
G1:$
&"#$'!L½
g /./ 2C2 1 =4ñ : < @& 6
I (−1) í' 8 ñ=ñ 4$ & ñBðBð‡ñò 2ð <9=4ñð ?:éï > & 8 )
ï 8 ñ : < & 6
8 ò Wð*):ñ,+.-•ð : :Ëñ < @& 6 í : < & 6
+ 6 <-=Óð 6 : $ îñµñ 6#" ñ 8
î ?4ñBð 3
ß»V!/"=Ø?Ï$±µ"/Ï?/50l1
'4:$yjË2$'
# (−1) êëS)S! ¶°\W$4W HqÒ0!'DæÑ!
Ó0!#$/
S
!:\ÌA?I$4±. '1Ë2$'(±Ø‹1S½
Y $'/.$ 2 2 ñ < M ñ 8 ï 8 -•ð 698" ?&î# && ñ'îÙð*):ñ,+.-0/[ñ & : 6 < 6 8 : < @& 6
8 ò
+ =4ñ & ñ ω 6 ð 8 ï 8 ñ & ï & ñ'î 8 ?: ñ & 3 1 =4ñ 8 det(A) =
)
A = [ω] + M,
−π
det(M ).
2 & ï‡ï >43É-ÒIÐÒËÍÌ −1 S®S!
W!#±#!ÊÑ'0!AÖ
D/"½‹É-#?'0!:D/"#Vl"
HqMV
²[ω]
/¢(4'V'0!:D/"#?ÙW/K$
ÏÌ
&"¶!2à
"0:2$µ4½V !
Ô":/'E!(s/'/y0!:D/"= M −1 [ω] /•½VJK (λ, x) H)Wu0!4'Ȁ!È$'=Ë2$ÍÌ)½[34
! M −1 [ω]x = λx
Ø l"#l4 x∗ [ω]x = λx∗ M x. É-µ
0!:È4#lÙÈ'FE
¼"
È'Ë0!4
E®È/'}¹4#Ë'²/•([²BG'¼Ê
½Ùê빀!//'wØ?
$42 [ω]x = 0, ±$4 0 = [ω]x = λM x. %& M 1!&Öë'0!"/(·'
I (−1)
. .
€!//'‡Ø? $42 λ = 0. É-€[//L0!:D/" M −1 [ω] 1Ê#½ !AÖ
ÔA"#:$/ÏEÓ$1!/ÏEG0!D/'"- I + M −1 [ω] 1. +È‡Ø Ø‹1²/'"/$
det(A) = det(M (I + M −1 [ω])) = det(M ) · 1
g /./ 2+2 ñ< n ñµñ ñ 8 3 1 =4ñ 8 det(I (−1) ) = det(S ).
n
n
2 & ï‡ï >43¼%&' S l
I\ØÖF
EAS\$
GÚDL(¹/'/Ù'0!:D/"#lA"l'
n Ø?
'0!Ò4 ±ib,
-
²/¢½ÙÉ-
€!
(4τÊW؋V±'0!:D/'"
S , '„Ø‹!"/'WH)?=
q²bS0:D/"½Ù%&"#BS0!:D/"Ù1ÙA"
H:EµÉ-n!ì<#½Ï6!½„É-\€!( S '0!"/#SÉ-#!
ì<#½¥<1/'
$®$#VÔ"4/ÏFE I (−1) = [ 1 ] + Sn 0!'D-$1'±"/ϲ½
2
Áq¿sÅ(
ó |¨š\¨=r„šœ•¨B¸²–© ‡›
’2³£–˜šsšB“뚦«©–©šjxf›w–©š\’w¨jš¦’2 )”w–¥{
sžf¨\’w§– ùC÷5ù,%Fú‹û !öB÷ ÷Íøwù
ýFÿ‡ú Lösý Bú‹ú-ù£÷ %FûÏü·ö'öyý\,ù "5ý Bú‹ú-ù£÷ %FûÏüLú-öB÷ %Fû …1šB“•œ¢–¥¯Ùp·’2š\«'vp·™w™w«v ó
ô\ô 
™v { v
šw–©¨ry–xf›C’w› ‡›£“¢“¢¨ !ù !!ù %¢ö Ü"û !ù '$# *'\ö¦÷'û &%(' % ,¥ù£ú‹ý?&ö ' ¦ú *!ÿ‡÷5ö¦÷'
û ÷Íøwù
) û%ù,+*Bö&Üÿ2ù£ý ,-yöB÷
û &
ö /. ù *)Üû 0÷ !LõVöB÷ %FûÏü•ù£1ý A–¥’w›š¦œAps«¥§2› wœëšKš¦’ n5“•—Aps™w™w«©–©³
šB“•–¥¨’2—C
3546 ó
ôô ²™v ó { ‡óô v
7 ry¨\Ÿw§«©š—4¼
v )«˜š¦œ -! 'K,ù %F÷'ÿ %,ëö¦÷'
û &ö )'Ùþû ú‹û ©&ö %Fû ÷ 8.ù *)Üû 9÷ !;: *2,ù %¢öB÷ &%ë1ý ƒy™›
œ•š¦“¢¨œ
tL”w›
¨\œ•¡=<pK w¸$š\’2³£›—Kš¦’2 p·™2™w«¥–˜³Cš¦“¢–©¨\’2—
+ >=?@ óô\ô ²™v 7 { 7 v
t&v ٚ¦–©«©š¦“¢”?š\’2 -p¼vw†‡š$¡›
:B A öBýëC
÷ -y!ù ¬ûÏö ,¥Eù D %&%Fû ÷Íø²ú‹ý !%sõVöB÷ %FûÏüëùFý "4û ÷ ø?þ÷ %Fÿ2üC÷'ÿ %•ù
)GF ÿ2öB÷
û &21ý †‡n€ps…®H
 q”w–¥«˜š\ w›C«©™w”w–˜I
š ó
ôô\ô v
|
v
A
w
Ÿ
2
’
[
¦
š
2
’
I
!
v
f
x
B
¨
)
ž
›
•
œ
—

þû ü‹õVù£÷Íø '$Jý !&%LKLÿ2ö '&%•öB÷ÿ %¢ù&ö ' û Ms,ù %¢,ù 2÷ûÏ&ö )GF ÿ2ö ²
†
€
n
y
p
®
…

+

2
”
¥
–
˜
«
\
š
‡
›
¥
«
2
™
w
”
©
–
š
ó
ô\
÷
û &2Ný ô v
|
\
¨
w
”
¶
’
1
…
š
2
”
\
¨
w
Ÿ
«

O
: h÷Íøwùù£û %!
ù 5*$ùëüC÷ %ëý ,[ý Bú ú-ù£÷ %FûÏP
ü . ù *¬û 0÷ !Wúö¦÷ %FûÏüëù£1ý n}~~~
t&œ•š\’2—•š\³F“•–¥¨’2—¼¨’SpK³£¨Ÿ2—°“¢–˜³C—
4†‡™›
›
³ë”š\’2 S†²–©§\’š¦Q« œ¢¨‡³C›
—•—°–©’w§2 35R ó
ô wSó ™v {
v
S œ•š\’ -=†²“¢›C’2§\›Cœ õVö¦÷ %FûÏüëù£ý ¼þû ü¹õVùC÷ ø 'BNý |\¨Ÿwœ¢’š¦«¨\Tª )¨¤™wŸ‡“•š¦“¢–©¨\’2š\«Aš¦’ jp·™‡{
™w«©–©›
=…1šB“•”w›C¤-š¦“¢–˜³C—
+ ? 4 ó
ô\ô ‡™v ¦ô { 7 ó v
ª )¨\¤™wŸw“•šB“•–¥¨’ 4 >
O œ•š\’ -ˆ“•›C’w§›Cœ ü•ö¦÷'û &% &+*&&¬ÿ‡÷'
û w1ý …VšB“¢”2›C¤-šB“•–©³
—-¨¦;
ó
ô\
ô ‡™v ‡óó { 7 v
ô œ•š\’ -҆“¢›
’w§\›
œ
V U ÿ‡ú,ù %FûÏüë&ö yõVù£÷Íø '$X
O
ý WsöBý£ù ' &þû ü®&ö )'YD &ö B÷'ûÏü A ÿ üC÷'
û w1ý †²™2œ¢–©’w§\›
œ
ws›Cž±o¨\œ -Z óô\ô 7 v
ó ºÓ–©«¥«©–˜š¦¤ v&t&œ¢›
’2³ë” )þ Bú-ùþ *2ùëü£ûÏö ' % *2!ù %F÷ûÏùFý \[‹!ù %Fú û ÷'ûÏö ]. ù *)Üû 0÷ !õ1ö¦÷ %FûÏüëù£1ý †²–˜š¦¤ |2vw…VšB“¢œ•–¥¯Èp·’2š\«'vp·™w™w«v©^ B_` ó
ôô ²™v ô 7 { ô v
) }ú-öBû ö ''%¢ù£ý•ý <bacd/ef&g5hc+i+j&kml
noHgqp&rs=hal,gqpd
) }ú-öBû ö ''%¢ù£ý•ý <utkSv+rqoHg+kiw=aqtvQlxd&s+yl,gqpqd