Error Controllability Under Finite-Order Discipline: PSWF/DPSS Extremal Windows Uniqueness, Integer Leading Terms (Spectral Flow/Index of Projection Pairs), and $10^{-3

E o Con ollabili y Unde Fini e-O de Discipline:
PSWF/DPSS Ex emal Windows Uniqueness, In ege Leading
Te ms (Spec al Flow/Index o P ojec ion Pai s), and
10−3
-Le el
Uni e sal Cons an s
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Unde unied Fou ie no maliza ion
b
(ξ) = RR ( )e−2πi ξ d
( equency in cycles), we con-
s uc e o discipline a ound ime-limi ingband-limi ing conca ena ed ope a o s: windowing
main leakage, mul iplica i e c oss- e ms, and sumin eg al die ence a e o ganized in o com-
pu able chains o  opological in ege leading e ms + analy ic ail e ms. On he con inuous
side,
Kc=DTBΩDT
(disc e e side
KN,W =TNBWTN
) yields uniqueness o ex emal win-
dows and leakage iden i y
|(I−BΩ)g∗|2
2= 1 −λ0
; on any undamen al domain o leng h 1,
squa ed-sum aliasing ene gy equals ou -o -domain ene gy; mul iplica i e c oss- e ms' Hankel-
ype blocks yield Hilbe Schmid (HS) exac o mulas; Eule Maclau in (EM) emainde an-
aly ic ail e ms a e con olled in closed o m by pe iodic Be noulli sup emum cons an s and
BPW inequali y. Based on explici non-asymp o ic eigen alue uppe bounds in na u al loga-
i hm calibe , we ob ain window-shape-independen minimal in ege Shannon numbe h esholds
(ε, N⋆
0) = (10−3,33),(10−6,42),(10−9,50)
. Unde denabili y hypo heses ( ace-class die ence
and s ongly con inuous pa hs), spec al ow equals index o p ojec ion pai s, iden i ying in ege
leading e ms o e o s as opological in a ian s. Comple e p oo s and ep oducible p ocedu es
a e p o ided.
Keywo ds
: Timeband limi ing; P ola e Sphe oidal Wa e Func ions (PSWF); Disc e e P o-
la e Sphe oidal Sequences (DPSS); Shannon numbe ; aliasing; Hankel block; Hilbe Schmid
no m; Eule Maclau in emainde ; spec al ow; index o p ojec ion pai s; de B anges
1 In oduc ion & His o ical Con ex
The ime-limi ingband-limi ing p oblem occupies a cen al posi ion in signal p ocessing and ha -
monic analysis. The SlepianLandauPollak amewo k e eals ha unde conca ena ed cons ain s
o ni e ime window
[−T, T]
and ni e bandwid h
[−Ω,Ω]
, wa e o ms wi h op imal ene gy concen-
a ion a e gi en by PSWF/DPSS, whose eigen alues clus e exponen ially nea
1
and
0
. Enginee -
ing p ac ice commonly encoun e s h ee ypes o e o swindowing ou -o -band leakage, aliasing,
and sumin eg al die ence (Poisson/EM emainde )his o ically ea ed sepa a ely, leading o
incompa able and non- ep oducible h esholds and cons an s.
This pape p oposes, unde unied cycles no maliza ion, a compu ablep o able ep oducible
e o discipline chain: (1) P ecisely cha ac e ize windowing main leakage ia p incipal eigen alue
λ0
o
Kc=DTBΩDT
(disc e e side
KN,W =TNBWTN
); (2) Reduce aliasing o ou -o -band en-
e gy ia he iden i y squa ed-sum aliasing = ou -o -domain ene gy; (3) P o ide HankelHS exac
o mulas and bounds o ou -o -band leakage a e mul iplica i e ac ion; (4) Cha ac e ize in ege
1
leading e ms ia he amewo k spec al ow = index o p ojec ion pai s, gi ing closed- o m
EM analy ic ail e ms ia Vaale Li mann ex ema and pe iodic Be noulli cons an s; (5) Gene -
a e
window-shape-independen
minimal in ege Shannon numbe h esholds ia explici non-
asymp o ic eigen alue uppe bounds in na u al loga i hm calibe . Thus, h ee blocks o e o s all
educe o h ee compu able quan i ies:
1−λ0
, HankelHS, and EM ail e ms; in ege leading e ms
a e ca ied by spec al in a ian s.
2 Model & Assump ions
Fou ie and uni s
:
b
(ξ) = RR ( )e−2πi ξ d
,
ξ
in cycles; Planche el:
| |2=|b
|2
.
P ojec ions
: Time-limi ing
DT =1[−T,T ]
; band-limi ing
BΩ =F−1(1[−Ω,Ω] b
)
.
Conca ena ed ope a o s
: Con inuous side
Kc=DTBΩDT
; disc e e side
KN,W =TNBWTN
(
TN
leng h-
N
es ic ion,
BW
band-limi ing p ojec ion o
[−W, W ]⊂[−1
2,1
2]
).
Shannon numbe
:
c=πTΩ
,
N0= 2TΩ = 2c/π
(con inuous);
N0= 2NW
(disc e e). Bo h
sides aligned ia
N0
.
BPW
: I
supp bg⊂[−Ω,Ω]
, hen
|g(m)|2≤(2πΩ)m|g|2
.
No ms
:
|·|2,|·|∞
; o any bounded ope a o
|A|op ≤ |A|HS
.
Fundamen al domain
: Any leng h-1 in e al
I= [a, a + 1)
, o aliasing ene gy no maliza ion.
3 Main Resul s (Theo ems and Alignmen s)
Theo em 1
(Theo em 1: Ex emal Window Uniqueness and Leakage Iden i y)
.
Le
Kc=DTBΩDT
be a compac sel -adjoin posi i e ope a o ac ing on
L2(R)
. I s la ges eigen alue
λ0∈(0,1)
is
simple, wi h co esponding eigen unc ion
g∗
(
|g∗|2= 1
) unique (up o phase), sa is ying
|BΩg∗|2
2=λ0,|(I−BΩ)g∗|2
2= 1 −λ0.
The disc e e side
KN,W =TNBWTN
is comple ely pa allel, wi h p incipal ec o being he  s
DPSS, also simple.
Alignmen 2
(Alignmen 1: Squa ed-Sum Aliasing = Ou -o -Domain Ene gy)
.
Fo any
w∈L2(R)
and any undamen al domain
I= [a, a + 1)
o leng h 1,
ZIX
k=0 bw(ξ+k)2dξ =ZR Ibw(η)2dη.
When
I
aligns wi h physical passband
[−Ω,Ω]
( ia ansla ion/scaling), he igh side is ou -o -
domain ene gy ela i e o ha passband; aking
w=g∗
yields
Aliasing(g∗;I)=1−λ0
.
Theo em 3
(Theo em 2: HS Exac Fo mula and Bounds o Mul iplica i e C oss-Te ms)
.
Fo
x∈L∞∩L2
and
W∈(0,1
2]
,
|(I−BW)MxBW|2
HS =ZR|bx(δ)|2σW(δ)dδ, σW(δ) = min(2W, |δ|).
Fu he mo e, o any
w∈L2
,
|(I−BW)Mxw|2≤ |(I−BW)MxBW|op|w|2+|x|∞|(I−BW)w|2,
2
hus o ex emal window
g∗
(Theo em 1):
|(I−BW)Mxg∗|2≤ |(I−BW)MxBW|HS +|x|∞p1−λ0.
Mo eo e ,
(I−BW)MxBW≡0
i and only i
x
is a.e. cons an .
Theo em 4
(Theo em 3: Analy ic Tail Bounds o EM Remainde )
.
Fo
g∈W2p,1(R)∩L2(R)
,
|R2p(g)| ≤ 2ζ(2p)
(2π)2p|g(2p)|L1.
I addi ionally
supp bg⊂[−Ω,Ω]
and local e alua ion on ime-domain leng h
L
,
|R2p(g)|
|g|2≤2ζ(2p)√LΩ2p.
Sucien h esholds achie ing
|R2p(g)|/|g|2≤10−3
a e
√LΩ4≤4.6197 ×10−4,√LΩ6≤4.9148 ×10−4,√LΩ8≤4.9797 ×10−4.
Theo em 5
(Theo em 4: Spec al Flow = Index o P ojec ion Pai s: Topologizing In ege Leading
Te ms)
.
Take smoo h equency mul iplie
ϕ∈C∞
c(R)
, le
Π = F−1MϕF
and o hogonal p ojec ion
P=1[1/2,∞)(Π)
. Le modula ion g oup
Uθ ( ) = e2πiθ ( )
,
Pθ=UθPU∗
θ
. I (i)
P−Pθ∈ S1
; (ii)
θ7→ Uθ
s ongly con inuous, hen sel -adjoin pa h
A(θ) = 2Pθ−I
admi s spec al ow, wi h
S A(θ)θ∈[θ0,θ1]= indP, Pθ1−indP, Pθ0∈Z.
Thus in ege leading e ms o sumin eg al die ence can be iden ied as ela i e indices along
pa hs; analy ic ail e ms con olled by Theo em 3.
Theo em 6
(Theo em 5: KRD Non-Asymp o ic P incipal Value Bound and Minimal In ege
Th esholds)
.
Le
N0= 2TΩ
(con inuous) o
N0= 2NW
(disc e e), hen he p incipal alue sa ises
1−λ0≤10 exp −(⌊N0⌋−7)2
π2log(50N0+ 25)!,
and dene he minimal in ege h eshold achie ing leakage bound
ε
:
N⋆
0(ε) := min nn∈N: 10 exp−(n−7)2
π2log(50n+25) ≤εo.
Nume ical alues (na u al loga i hm, oo in exponen ):
(ε, N⋆
0, c⋆, NW⋆) = (10−3,33,π
2·33,16.5),(10−6,42,π
2·42,21.0),(10−9,50,π
2·50,25.0).
4 P oo s
4.1 P oo o Theo em 1
Compac ness and sel -adjoin ness
.
BΩ
and
DT
a e o hogonal p ojec ions,
(DTBΩ)( , s) =
1[−T,T ]( )sin(2πΩ( −s))
π( −s)
is squa e-in eg able on
[−T, T]2
, so
DTBΩ
is Hilbe Schmid , hus
Kc=
DTBΩDT
is compac sel -adjoin .
3
Commu a i i y and simplici y
. Le
x= /T ∈[−1,1]
,
c=πTΩ
. Classical PSWF sa ises
(1 −x2)y′′(x)−2xy′(x)+(χ−c2x2)y(x)=0.
W i e
Lcy:= −d
dx(1 −x2)dy
dx+c2x2y
, e o mula ing as
Lcy=χy
.
Lc
is a sel -adjoin S u m
Liou ille ope a o on
[−1,1]
, wi h endpoin s being egula singula poin s, spec um pu ely disc e e
wi h each eigen alue simple; eigen unc ions' ze o coun s ma ch hei indices (oscilla ion heo em).
Slepian commu a i i y shows
Kc
and
Lc
can be simul aneously diagonalized, so geome ic mul i-
plici y o
λ0
equals mul iplici y o co esponding
χ0
, hence 1, wi h p incipal eigen unc ion unique
(up o phase).
Leakage iden i y
. By o hogonal p ojec ion p ope y o
BΩ
and
|g∗|2= 1
,
λ0=⟨Kcg∗, g∗⟩=⟨BΩg∗, g∗⟩=|BΩg∗|2
2,|(I−BΩ)g∗|2
2= 1 −λ0.
Disc e e side
KN,W
wi h commu ing second-o de die ence ope a o o ms disc e e S u m heo y,
p incipal alue also simple.
4.2 P oo o Alignmen 1
R I=Fk=0(I+k)
is a coun able disjoin decomposi ion. Since
bw∈L2
,
Pk=0 RI|bw(ξ+k)|2dξ ≤
|bw|2
2<∞
, Tonelli applies. Va iable subs i u ion
η=ξ+k
yields
ZIX
k=0 |bw(ξ+k)|2dξ =X
k=0 ZI+k|bw(η)|2dη =ZR I|bw(η)|2dη.
4.3 P oo o Theo em 2
HS exac o mula
. F equency-domain ke nel
K(ξ, η) = 1|ξ|>W 1|η|≤Wbx(ξ−η).
HS no m squa ed
ZZ |K|2=Z|η|≤WZ|ξ|>W |bx(ξ−η)|2dξdη =ZR|bx(δ)|2mW(δ)dδ,
whe e
mW(δ) = meas{η∈[−W, W ] : |η+δ|> W }
. Geome ically he measu e o complemen -
in e sec ion be ween leng h-
2W
in e al and i s ansla ion:
mW(δ) = min(2W, |δ|)
. This yields he
s a ed o mula.
Bounds
. Decompose
(I−BW)Mx= (I−BW)MxBW+ (I−BW)Mx(I−BW),
apply
|A|op ≤ |A|HS
,
|(I−BW)| ≤ 1
and iangle inequali y o ob ain gene al bound; aking
w=g∗
and using Theo em 1's
|(I−BW)g∗|2≤√1−λ0
yields s a ed o mula. I
(I−BW)MxBW≡0
, hen
o any band-limi ed inpu
BWw
,
bx∗(bw·1[−W,W ])
suppo emains in
[−W, W ]
, o cing
supp bx⊂ {0}
;
combined wi h
x∈L∞
lea es only cons an unc ions.
4
4.4 P oo o Theo em 3
Pe iodic Be noulli's Fou ie expansion gi es

B2p(·)
(2p)! ∞
=2ζ(2p)
(2π)2p.
EM emainde o mula
R2p(g) = ZR
g(2p)( )B2p({ })
(2p)! d ,
hus
|R2p(g)| ≤ 2ζ(2p)
(2π)2p|g(2p)|L1.
I
supp bg⊂[−Ω,Ω]
, hen
|g(2p)|L1≤√L|g(2p)|L2≤√L(2πΩ)2p|g|2
. The
(2π)2p
in denomina o and
BPW's
(2π)2p
comple ely cancel, yielding s a ed o mula and h eshold alues.
4.5 P oo o Theo em 4
Dene
A(θ)=2Pθ−I
. By hypo heses (i)(ii) and egula i y o spec al p ojec ions,
A(θ)
is a
s ongly con inuous pa h o sel -adjoin F edholm ope a o s. Spec al ow is dened as signed
coun o ze o c ossings; on he o he hand, ela i e index
ind(P, Q)
when
P−Q∈ S1
can be
dened ia ela i e dimension, sa is ying addi i i y and homo opy in a iance. Subdi ide
[θ0, θ1]
in o small in e als making
0
a egula alue on each segmen ; locally, spec al ow equals jumps in
ank(Pθ| anP)
; ela i e index also eco ds he same jumps. Conca ena e and use homo opy in a iance
o ob ain
S (A(θ))θ1
θ0= ind(P, Pθ1)−ind(P, Pθ0).
F equency-domain smoo hing
ϕ∈C∞
c
and (when necessa y) ime-domain localiza ion ensu e
P−
Pθ∈ S1
; in s ong ope a o opology le
ϕ→1[−W,W ]
, in ege in a ian , hus es ablishing opological
o igin o in ege leading e ms.
4.6 P oo o Theo em 5
Disc e e-side o iginal o mula wi h
NW
as pa ame e :
1−λ0≤10 exp−(⌊2NW⌋−7)2
π2log(100NW + 25).
Se ing
N0= 2NW
yields
1−λ0≤10 exp−(⌊N0⌋−7)2
π2log(50N0+ 25).
Con inuous-side bound exp essed in
c
ia
N0= 2c/π
gi es same unied o m. Fo gi en
ε
, scan
minimal in ege
n
such ha igh side
≤ε
o ob ain
N⋆
0(ε)
. Nume ical alues as s a ed.
5 Model Applica ions
Con inuousdisc e e mapping
:
(T, Ω) ↔(N, W )
aligned ia
N0= 2TΩ=2NW
; ypically ake
N≈2T
,
W≈Ω
(when uni s consis en ).
5

Consis ency o wo calibe s
:
DTBΩDT
and
BΩDTBΩ
ha e iden ical non-ze o spec a (p opo-
si ion:
AB
and
BA
ha e iden ical non-ze o spec a). F equency-domain leakage:
|(I−BΩ)g∗|2
2=
1−λ0
; ime-domain leakage:
|(I−DT)w∗|2
2= 1 −λ0
.
Fundamen al domain consis ency
: Choose undamen al domain consis en wi h
[−Ω,Ω]
( ansla ion/scaling aligned), Alignmen 1's igh side is ou -o -domain ene gy ela i e o ha
passband, hus o
w=g∗
, aliasing ene gy equals
1−λ0
.
6 Enginee ing P oposals
(1) Th eshold-d i en pa ame e selec ion
Gi en leakage ole ance
ε∈ {10−3,10−6,10−9}
,
consul Theo em 5 o minimal in ege
N⋆
0
, acco dingly se
(T, Ω)
o
(N, W )
.
(2) Compu able bounds o c oss- e ms
One FFT ob ains
bx
, compu e
ΞW(x) := ZR|bx(δ)|2min(2W, |δ|)dδ1/2
.
Budge o mula
|(I−BW)Mxg∗|2≤ΞW(x) + |x|∞p1−λ0.
I
bx
p e-l e ed and na owband,
ΞW(x)
signican ly educed.
(3) EM o de selec ion
Gi en
(L, Ω)
, choose smalles
p∈ {2,3,4}
such ha
√LΩ2p≤
10−3/(2ζ(2p))
.
(4) Mul i- ape /mul i-passband
Take  s
K≈ ⌊N0⌋
DPSS o mul i- ape ; aliasing budge
along Alignmen 1 accumula es pe ape , c oss- e ms es ima ed blockwise pe Theo em 2 by
bx
's
ene gy dis ibu ion.
7 Discussion (Risks, Bounda ies, Pas Wo k)
Denabili y bounda ies
: Spec al ow = index o p ojec ion pai s elies on
P−Pθ∈ S1
.
Sha p band-limi ing p ojec ion and pu e modula ion die ence gene ally non- ace-class, equi -
ing equency-domain smoo hing  s and (when necessa y) ime-domain localiza ion, hen aking
spec al p ojec ion, nally app oaching limi in s ong ope a o opology; in ege leading e ms
insensi i e o egula iza ion de ails.
Scales and cons an s
: Unde cycles no maliza ion, BPW's
(2π)m
and EM cons an denomi-
na o
(2π)2p
comple ely cancel; KRD h eshold exp essed in na u al loga i hm wi h
log(50N0+25)
,
oo in exponen yields minimal in ege .
Conse a i e s igh
: Can gene a e h esholds pe Theo em 5's igh e sion (
33,42,50
), o
unde ex eme isk a e sion choose la ge in ege s, o ming conse a i e edundancy.
His o ical h eads
: Timeband ex ema, Toepli z index/winding numbe , spec al ow/ ela i e
index, and one-sided ex ema cons i u e heo e ical backbone; non-asymp o ic h esholds connec
classical asymp o ics wi h enginee ing pa ame e iza ion.
8 Conclusion
Unde unied no maliza ion and pa ame e mapping, h ee e o ypesmain leakage, mul iplica-
i e c oss- e ms, and sumin eg al die encea e inco po a ed in o ope a o - heo e ic discipline o
in ege leading e ms + analy ic ail e ms:
6

Main leakage
p ecisely cha ac e ized by
λ0
, wi h explici non-asymp o ic bounds gene a ing
window-shape-independen
minimal in ege h esholds;

C oss- e ms
quan ied ia HankelHS exac o mulas, yielding compu able bounds wi hou
heu is ic cons an s;

EM emainde
unde cycles no maliza ion exhibi s 
(2π)
comple e cancella ion, wi h closed-
o m cons an s di ec ly in e acing imeband pa ame e s;

In ege leading e ms
(spec al ow/index o p ojec ion pai s) p o ide opological o igin.
The esul ing ni e-o de discipline achie es bo h enginee ing implemen a ion and comple e
ma hema ical ancho ing.
A No a ion, Uni s, and Basic Tools

No maliza ion:
b
(ξ) = RR ( )e−2πi ξ d
,
ξ
in cycles.

P ojec ions:
DT =1[−T,T ]
,
BΩ=F−11[−Ω,Ω]F
.

No ms:
|A|op ≤ |A|HS
, Planche el:
| |2=|b
|2
.

Shannon numbe :
N0= 2TΩ=2NW
.
B
AB
and
BA
Ha e Iden ical Non-Ze o Spec a
I
ABx =λx
wi h
λ= 0
, hen
Bx = 0
and
BA(Bx) = λ(Bx)
; e e se di ec ion simila . Thus
DTBΩDT
and
BΩDTBΩ
ha e iden ical non-ze o spec a; leakage iden i ies in bo h calibe s a e
equi alen .
C PSWF S u mLiou ille S uc u e and P incipal Value Simplici y
Va iable
x= /T ∈[−1,1]
,
c=πTΩ
. PSWF sa ises
(1 −x2)y′′(x)−2xy′(x)+(χ−c2x2)y(x)=0.
W i e
Lcy:= −d
dx(1 −x2)dy
dx+c2x2y,
hen
Lc
is sel -adjoin second-o de die en ial ope a o on
[−1,1]
. Endpoin s
x=±1
a e egula
singula poin s, spec um pu ely disc e e wi h each eigen alue simple; eigen unc ions' ze o coun s
ma ch indices. Slepian commu a i i y shows
Kc
and
Lc
sha e o hogonal eigensys em, so
Kc
's
p incipal eigen alue has geome ic mul iplici y 1.
Disc e e side
: Toepli z- ype p ola e ma ix commu es wi h idiagonal die ence ope a o ;
disc e e S u m oscilla ion heo em gua an ees p incipal alue simplici y.
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D Squa ed-Sum Aliasing = Ou -o -Domain Ene gy De ails
Fo
I= [a, a + 1)
, we ha e
R I=Fk=0(I+k)
(disjoin ). Fo any
w∈L2
,
X
k=0 ZI|bw(ξ+k)|2dξ =X
k=0 ZI+k|bw(η)|2dη =ZR I|bw(η)|2dη.
Squa ed-sum means aking modulus squa ed o each ansla ed channel be o e summing p io o
in eg a ion; dis inguished om enginee ing measu es o pe iodize  s hen ake modulus squa ed
(which con ain c oss- e ms).
E HankelHS Geome ic Measu e Piecewise Calcula ion
Fo xed
δ
, he se
S(δ) = {η∈[−W, W ] : |η+δ|> W}.
I
|δ| ≤ 2W
, hen
S(δ)
is union o wo endpoin pieces each o leng h
|δ|/2
, measu e
|δ|
; i
|δ|>2W
,
hen
S(δ)=[−W, W]
en i ely, measu e
2W
. Thus
mW(δ) = meas S(δ) = min(2W, |δ|),
yielding Theo em 2's HS exac o mula.
F EM Remainde and 
(2π)
Cancella ion De ails
Pe iodic Be noulli sup cons an

B2p(·)
(2p)! ∞
=2ζ(2p)
(2π)2p.
I
supp bg⊂[−Ω,Ω]
, hen
|g(2p)|L1≤√L|g(2p)|L2≤√L(2πΩ)2p|g|2,
subs i u ing in o EM emainde bound, denomina o
(2π)2p
and BPW's
(2π)2p
comple ely cancel,
ob aining
|R2p(g)|
|g|2≤2ζ(2p)√LΩ2p.
G Rep oducible Checklis (Pseudocode)
KRD h eshold (na u al loga i hm)
de N0_s a (eps):
n = 1
while T ue:
U = 10*exp(- (n-7)**2 / (pi**2*log(50*n + 25)) )
i U <= eps:
e u n n, (pi*n/2), (n/2) # (N0*, c*, NW*)
n += 1
HankelHS c oss- e m
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# xha : Fou ie samples on g id del a wi h spacing ddel a
XiW_sq = sum( abs(xha )**2 * minimum(2*W, abs(del a)) ) * ddel a
XiW = sq (XiW_sq)
EM emainde h eshold
# choose smalles p in {2,3,4} wi h
# sq (L) * Omega**(2*p) <= 1e-3/(2*ze a(2*p))
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