PERMISSION TO ARCHIVE PUBLISHED CONTENT
Dea ,
We in o m o you ha idUS is he Resea ch Reposi o y o he Uni e sidad de Se illa, which lis s
he p oduc s o esea ch ac i i ies o his ins i u ion ollowing he Ins i u ional Decla a ion o
p omo e Open Access.
We con ac o you in o de o ask o pe mission o a chi e in idUS unde licence CC (by-nc-nd),
he ollowing wo k ealized by a membe o he Uni e sidad de Se illa and published wi h you.
A. Au ho
Name, su name: Mi a Ma ía Cas o Smi no a
Email: [email p o ec ed] Phone:
B. Wo ks o be deposi ed
Ti le: Time-and-band limi ing o ma ix- alued o hogonal polynomials ela ed wi h 2 × 2
hype geome ic ope a o s
DOI, ISBN...: h ps://doi.o g/10.1090/conm/807/16164
SOURCE, URL: Con empo a y Ma hema ics, Volume 807, 2024
C. Edi o ’s au ho iza ion o a chi e in idUS
Place and da e o sign Place and da e o sign
Se ille, Janua y 22 h, 2025 P o idence, Janua y 22, 2025
Au ho ’ sign On behal o Edi o ial Sign:
Name and posi ion
B idge Manzella, Associa e
Edi o o P oceedings
Time-and-band limi ing o ma ix- alued o hogonal
polynomials ela ed wi h 2×2hype geome ic ope a o s
M.M. Cas o, A. Foulqui´e-Mo eno, and A. F adi
Abs ac . We conside a amily o ma ix- alued o hogonal polynomials
o size 2 ×2, which a e common eigen unc ions o a di e en ial ope a o
o hype geome ic ype, in connec ion wi h he p oblem o ime-and-band
limi ing. The p oblem in ques ion is as ollows: gi en a global ope a o , de ined
by an in eg al ke nel o a ull ma ix, one looks o a local ope a o gi en by
a second o de di e en ial ope a o o a block idiagonal ma ix espec i ely,
commu ing wi h he global one.
We apply he echniques o he cons uc ion o he commu ing local
ope a o in oduced in a p e ious wo k by A. G ¨unbaum, I. Pacha oni and I.
Zu ian in he ma ix- alued se ing, whe e he ole o bispec ali y is c ucial.
This cons i u es an illus a i e example o an ex ension o a si ua ion in ol ing
ma ix o hogonali y, o a scala esul ha o igina es in a se ies o pape s by
D. Slepian, H. Landau and H. Pollak a Bell Labs in he six ies.
1. In oduc ion
The ime-and-band-limi ing p oblem has i s o igins in he wo k o D. Slepian,
H. Landau and H. Pollak a he Bell Labs back in he 1960’s, [30, 31, 39, 40, 41,
42, 43]. This se ies o pape s was mo i a ed by a ques ion posed by C. Shannon
in [38], o wha is he bes use one can do o he alues o he Fou ie ans o m
F (k) o , o alues o kin he band [−W,W] when ( ) is a ime-limi ed signal
wi h ini e suppo [−T ,T].
The nume ically s able econs uc ion o om his noisy da a leads o he
s udy o an in eg al ope a o in L2(−T ,T), gi en by means o he in eg al ke nel
k( , s) = sin W( −s)/( −s). One a i es o he p oblem o compu ing nume ically
mos o he eigen unc ions o his ope a o , which is a e y ill-condi ioned one,
since many o he eigen alues a e e y close oge he . In his sense, D. Slepian,
2020 Ma hema ics Subjec Classi ica ion. 33C45, 22E45, 33C47.
Key wo ds and ph ases. Time-band limi ing, Ma ix alued o hogonal polynomials.
The esea ch o he i s au ho was pa ially suppo ed by PID2021-124332NB-C21
(FEDER(EU) / Minis e io de Ciencia e Inno aci´on-Agencia Es a al de In es igaci´on) and FQM-
262 (Jun a de Andaluc´ıa).
The esea ch o he second and hi d au ho was suppo ed by Cen e o Resea ch and
De elopmen in Ma hema ics and Applica ions (CIDMA) om Uni e si y o A ei o h ough
p ojec s DOI: 10.54499/UIDB/04106/2020 and DOI: 10.54499/UIDP/04106/2020.
The hi d au ho would also like o exp ess his since e g a i ude o he Ma hema ical Physics
Labo a o y, Special Func ions and Applica ions (MaPSFA) o hei inancial suppo .
1
2 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
H. Landau and H. Pollak ound ha an app op ia e sel adjoin ex ension o he
ope a o
(e
D )(x) = ((T2−x2) 0(x))0−W2x2
commu es wi h he gi en in eg al ope a o and has simple spec um. Thus, he
eigen unc ions o bo h ope a o s a e he same.
In ce ain a eas o Ma hema ics, as in his p oblem, one a i es a a global
ope a o , gi en by an in eg al ke nel o a ull ma ix. In ce ain excep ional cases
one can exhibi a di e en ial ope a o o low o de o a ma ix wi h a small numbe
o diagonals (a local ope a o ) which has he same eigen unc ions as he global one.
Thus, he nume ical compu a ion o he eigen unc ions o he local ope a o , is
nume ically s able and much easie han compu ing nume ically he eigen unc ions
o he ini ial global ope a o in an accu a e way (see [34] o mo e de ails on
compu a ional issues).
In his pape we a e conside ing a noncommu a i e ex ension o he ime-
and-band limi ing p oblem desc ibed abo e. We display one illus a i e example,
in ol ing ma ix o hogonali y, whe e his p ope y holds. This ques ion was
al eady aken up in he ma ix- alued se ing in a ew e e ences such as [3, 4,
5, 6, 24, 25, 26], and also o he scala case in [7, 16, 17, 18, 19, 21]. This
p oblem has also impo an connec ions wi h Random Ma ix Theo y as one can
see in [33, 45, 46].
Speci ically, we deal wi h a amily o ma ix- alued polynomials (P(α,β, )
n)n≥0
o size 2 ×2, in oduced in [2], o hogonal wi h espec o he weigh ma ix
W(α,β, )gi en in (2.7) and (2.8). These polynomials a e common eigen unc ions o
a di e en ial ope a o o hype geome ic ype (in he sense de ined by A. Ti ao
in [44]), wi h diagonal ma ix eigen alues, wi h no epe i ion in hei en ies.
This amily cons i u es a new in e es ing example, among se e al o he s in he
ecen li e a u e, o ma ix- alued o hogonal polynomials sa is ying second o de
di e en ial equa ions, o in o he wo ds, exhibi ing a bispec al si ua ion acco ding
o [11].
The subjec o ma ix o hogonali y, s a ed by K ein in [28] and [29], has
ecei ed a lo o a en ion in he las wo decades in connec ion wi h di e en ial
equa ions a e he seminal wo k o A. Du ´an in [9] (see o ins ance [12, 13, 14,
20, 22, 23, 27, 35, 36, 37]).
The ole o bispec ali y appea s o be c ucial in he cons uc ion o he ime-
and-band limi ing commu ing ope a o , see [16] o classical o hogonal polynomials
and [26] o ma ix- alued ones. We apply he echniques in [26] o he cons uc ion
o he commu ing local ope a o s o he case we a e conside ing he e.
The con en s o he p esen pape a e as ollows. In he nex sec ion, we begin
wi h a gene al o e iew o ma ix o hogonali y and subsequen ly we in oduce he
amily o polynomials (P(α,β, )
n)n≥0. In pa icula , in [1] i was shown ha he
sequence o de i a i es o k- h o de , k≥1, o he polynomials (P(α,β, )
n)n≥0is also
o hogonal, a ac ha does no always hold ue in he ma ix- alued amewo k
(see [14, Sec ion 7]). In his case one has a Pea son ype equa ion (see [1, Theo em
5.4] and (2.18) below), which implies he o hogonali y o he de i a i es.
As i was shown in [8], analogously o he scala si ua ion, o a gi en amily o
ma ix- alued o hogonal polynomials (Pn)n≥0, he o hogonali y o he de i a i es
TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 3
is also equi alen o a linea ela ion be ween he polynomials Pn,P0
n,P0
n−1and
P0
n+1 (see [32]). We show his p ope y o he amily o polynomials (P(α,β, )
n)n≥0
conside ed he e, as one may see in Subsec ion 2.3.
In Sec ion 3 we gi e a gene al se up o he ime-and-band limi ing p oblem
in he ma ix- alued amewo k. The e o e, in Sec ion 4, we conside he amily o
polynomials (P(α,β, )
n)n≥0and i s sequence o de i a i es (P(α,β, ,k)
n)n≥kin connec ion
wi h his p oblem. Ou main ool o he cons uc ion o he commu ing di e en ial
ope a o is he P ope y M hypo hesis (see (4.1) below) aised and exploi ed in
[26]. In Sec ion 5, we conside he case when he global ope a o is gi en by a
ull ma ix, and cons uc a commu ing idiagonal banded ma ix o he example
a hand. Finally, in he line o [4, Sec ion 6] and [7, Sec ion 7], we display he
esul s o some nume ical compu a ions, showing he ad an ages o ha ing such a
commu ing local ma ix.
2. P elimina ies
2.1. Ma ix- alued o hogonal polynomials. We s a by gi ing some
backg ound on ma ix- alued o hogonal polynomials (see [15] o u he de ails).
A weigh ma ix Wis a complex R×Rma ix- alued in eg able unc ion on he
in e al (a, b), such ha Wis posi i e de ini e almos e e ywhe e and wi h ini e
momen s o all o de s, i.e., Rb
axndW(x)∈CR×R, n ∈N. The weigh ma ix W
induces a He mi ian sesquilinea o m,
hP, QiW=Zb
a
P(x)W(x)Q∗(x)dx,
o any pai o R×Rma ix- alued unc ions P(x) and Q(x), whe e Q∗(x) deno es
he conjuga e anspose o Q(x).
We conside a sequence (Pn)n≥0o o hogonal polynomials wi h espec o a
weigh ma ix W, whe e
Pn(x) = xnCn+xn−1Cn−1+···+C0, Ci∈CR×R, i = 0,1, . . . , n,
is a ma ix- alued polynomial wi h non-singula leading coe icien Cn, and hPn, PmiW
= ∆nδn,m, whe e ∆n,n≥0, is a posi i e de ini e ma ix. When ∆n=I, he e I
deno es he iden i y ma ix, we say ha he polynomials (Pn)n≥0a e o hono mal.
In pa icula , when he leading coe icien o Pn(x), n≥0, is he iden i y ma ix,
we say ha he polynomials (Pn)n≥0a e monic.
Gi en a weigh ma ix W, he e exis s a unique sequence o monic o hogonal
polynomials (Pn)n≥0in CR×R[x], any o he sequence o o hogonal polynomials
(Qn)n≥0can be w i en as Qn(x) = KnPn(x) o some non-singula ma ix Kn.
Any sequence o monic o hogonal ma ix- alued polynomials (Pn)n≥0sa is ies
a h ee- e m ecu ence ela ion
(2.1) xPn(x) = Pn+1(x) + BnPn(x) + AnPn−1(x), o n∈N0,
whe e P−1(x) = 0, P0(x) = I. The R×Rma ix coe icien s Anand Bnenjoy
ce ain p ope ies, in pa icula , Anis non-singula o any n.
Le Dbe a igh -hand side o dina y di e en ial ope a o wi h ma ix alued
polynomial coe icien s,
(2.2) D=
s
X
i=0
∂iFi(x), ∂i=di
dxi.
4 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
The ope a o Dac s on a polynomial unc ion P(x) as P D =Ps
i=0 ∂iPFi(x).
We say ha he di e en ial ope a o Dis symme ic wi h espec o Wi
(2.3) hPD, QiW=hP, QDiW, o all P, Q ∈CR×R[x].
A necessa y and su icien condi ion o a ma ix- alued di e en ial ope a o o be
symme ic is gi en in [12, Theo em 3.1]. Acco ding o his esul , he di e en ial
ope a o D=d2
dx2F2(x) + d
dxF1(x) + F0is symme ic wi h espec o Wi and
only i
F2W=WF∗
2,(2.4)
2 (F2W)0=F1W+WF∗
1,
(F2W)00 −(F1W)0=WF∗
0−F0W,
and
(2.5) lim
x→a,b F2(x)W(x) = 0 and lim
x→a,b (F1(x)W(x)−W(x)F∗
1(x)) = 0.
A amily o ma ix- alued o hogonal polynomials (Pn)n≥0sa is ying second
o de di e en ial equa ions, gi es a basis o polynomials in CR×R[x] ha a e join
eigen unc ions o a ixed di e en ial ope a o Do o de wo, as gi en in (2.2), wi h
ma ix- alued eigen alue Λn
(2.6) DPn(x)=ΛnPn(x).
A he same ime, hey a e eigen unc ions o a second o de di e ence ope a o
B, gi en by an in ini e idiagonal block-ma ix whose en ies a e gi en by he
coe icien s o he h ee- e m ecu ence ela ion in (2.1), wi h eigen alue θ(x) = x,
ha is o say,
BPn=θ(x)Pn.
In his sense, one says ha he amily (Pn)n≥0is bispec al, in he spi i o [11].
2.2. The amily o ma ix- alued o hogonal polynomials. In [2] he
au ho s in oduce a Jacobi ype weigh ma ix W(α,β, )(x) and a di e en ial ope a o
D(α,β, )such ha D(α,β, )is symme ic wi h espec o he weigh ma ix W(α,β, )(x).
Le α,β, ∈R,α, β > −1 and |α−β|<| |< α +β+ 2. We conside he
weigh ma ix unc ion
(2.7) W(α,β, )(x) = xα(1 −x)β
W(α,β, )(x), o x∈(0,1),
wi h
W(α,β, )(x) =
(2.8)




(κ ,β + 2)
κ ,−β
x2−(κ ,β + 2) x+ (α+ 1) (α+β+ 2)x−(α+ 1)
(α+β+ 2)x−(α+ 1) − (κ− ,β + 2)
κ− ,−β
x2−(κ− ,β + 2) x+ (α+ 1)



,
whe e, o he sake o clea ness in he es o he pape , we use he no a ion:
(2.9) κ± ,±β=α± ±β .
W(α,β, )is an i educible weigh ma ix and he hype geome ic ype di e en ial
ope a o gi en by

TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 5
(2.10) D(α,β, )=d2
dx2F2(x) + d
d F1(x) + F0(x),
whe e
(2.11) F2(x) = x(1 −x), F1(x) = C∗−xU and F0(x) = −V,
and
(2.12)
C=
α+ 1 −κ− ,−β
κ ,−β
−κ− ,−β
α+1+κ ,−β

, U = (α+β+ 4) Iand V= 0
0 0,
is symme ic wi h espec o W(α,β, ).
As shown in [2, Theo em 4.3], he sequence o ma ix- alued monic o hogonal
polynomials P(α,β, )
nn≥0associa ed wi h he weigh unc ion W(α,β, )(x) a e
common eigen unc ions o he di e en ial ope a o D(α,β, )wi h diagonal eigen alues
(2.13) Λ(α,β, )
n=λn0
0µn,λn=−n(n−1) −n(α+β+ 4) − ,
µn=−n(n−1) −n(α+β+ 4) .
Conside now he weigh ma ix gi en by
(2.14) W(k)(x) = W(α+k,β+k, )(x)
and le dk
dxkP(α,β, )
n(x) be he de i a i e o o de ko he monic polynomial P(α,β, )
n(x),
o n≥k. Then
(2.15) P(α,β, ,k)
n(x) = (n−k)!
n!
dk
dxkP(α,β, )
n(x)
a e monic polynomials o deg ee n−k o all n≥k. Obse e ha o he case k= 0
one has he amily o polynomials de ined ini ially by he weigh W(α,β, )(x) in
(2.7).
Conside he ope a o
(2.16) D(k)=D(α,β, ,k)=d2
dx2x(1 −x) + d
d ((C(k))∗−xU(k))−V,
wi h
C(k)=C+kI, U(k)=U+ 2kI = (α+β+ 4 + 2k) I,
As shown in [1, Sec ion 5], he sequence o de i a i es a e common eigen unc ions
o he hype geome ic ype ope a o D(k)wi h diagonal ma ix eigen alues
(2.17)
Λ(k)
n= Λ(α,β, ,k)= λ(k)
n0
0µ(k)
n!,λ(k)
n=−(n−k) (α+β+3+n+k)− ,
µ(k)
n=−(n−k) (α+β+3+n+k).
6 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
Mo eo e , he au ho s p o e ha he sequence o de i a i es P(α,β, ,k)
n(x) a e
o hogonal wi h espec o he weigh ma ix W(k)(x), which is a consequence o a
Pea son ype equa ion:
W(k)(x) Φ(k)(x)0=W(k)(x) Ψ(k)(x), k ∈N,wi h(2.18)
Φ(k)(x) = Ak
2x2+Ak
1x+Ak
0and Ψ(k)(x) = Bk
1x+Bk
0,
whe e
Ak
2=


−κ ,β + 2(k+ 2)
κ ,β + 2(k+ 1) 0
0−κ− ,β + 2(k+ 2)
κ− ,β + 2(k+ 1)



,
(2.19)
Ak
1=2
(κ− ,β + 2(k+ 1))(κ ,β + 2(k+ 1)) 0κ ,−β
κ− ,−β0−Ak
2,
(2.20)
Ak
0=κ ,−βκ− ,−β
(κ− ,β + 2(k+ 1))(κ ,β + 2(k+ 1)) −1 1
−1 1,
(2.21)
Bk
1= (α+β+ 4 + 2k)Ak
2,
(2.22)
Bk
0=−(α+k+ 1)I−1
−κ− ,−β0
0κ ,−βAk
2
(2.23)
+1
2 α+β+ 2k+ 4
Ak
1+Bk
1−κ− ,β −2(k+ 1) 0
0κ ,β + 2(k+ 1).
Pa icula ly, one has ha
(2.24) W(k+1) =W(k)Φ(k)(x),
whe e W(k+1) is p ecisely he o hogonali y weigh associa ed o he sequence o
de i a i es o o de k+ 1. This ela ion allows o cons uc symme ic di e en ial
ope a o s wi h espec o he weigh W(k)using he en ies o he equa ion. Indeed,
in his case one has ha he di e en ial ope a o (see [1, Co olla y 6.4])
(2.25) E(k)=d2
dx2(Φ(k)(x))∗+d
dx(Ψ(k)(x))∗
is symme ic wi h espec o W(k)(x) o all k∈N0.Mo eo e , he polynomials
P(α,β, ,k)
n+kn≥0a e eigen unc ions o he ope a o E(k)wi h eigen alue
(2.26) ΛnE(k)=n(n+α+β+ 3 + 2k)Ak
2.
Thus, o each ixed alue o k≥0, we ha e wo di e en symme ic ope a o s
D(k)and E(k)in he algeb a o ma ix di e en ial ope a o s ha ing as common
eigen unc ions he sequence o polynomials P(α,β, ,k)
n(x)n≥0, o hogonal wi h espec
TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 7
o he weigh W=W(k). This ac will be e y use ul o cons uc commu ing
di e en ial ope a o s in he ime-and-band-limi ing amewo k below in sec ion 4.
In he nex sec ion we comple e he cha ac e iza ion o he p ope y o o hogonali y
o he sequence o de i a i es P(α,β, ,k)
n(x)n≥k o he amily o ma ix- alued
polynomials P(α,β, )
n(x) conside ed he e.
2.3. A cha ac e iza ion p ope y o he o hogonali y o he de i a i es.
Following he esul s in [32] o classical o hogonal polynomials and he esul in [8,
Theo em 3.14] we show he ollowing ecu ence ela ion in ol ing he polynomials
P(α,β, ,k)
n(x) and he sequence o hei de i a i es, which a e a he same ime
o hogonal ma ix- alued polynomials wi h espec o he weigh ma ix W(k+1) in
(2.14).
P oposi ion 2.1.The monic o hogonal polynomials (P(α,β, ,k)
n(x))n≥kand
hei de i a i es sa is y he ecu ence ela ion
P(α,β, ,k)
n(x) = 1
n+ 1 −kP(α,β, ,k)
n+1 (x)0
+B(α+k,β+k, )
n−k−B(α+k+1,β+k+1, )
n−1−kP(α,β, ,k)
n(x)0
−1
α+β+k+n+ 2A(α+k,β+k, )
n−kP(α,β, ,k)
n−1(x)0,
whe e, o n≥0, he exp essions o he coe icien s A(α,β, )
na e (see [1, P oposi ion
2.3] and [2, Theo em 3.12])
A(α,β, )
n=a(α,β, )
n(4 + 2n+κ− ,β) (2n+κ ,β) 0
0 (4 + 2n+κ ,β ) (2n+κ− ,β ),
wi h a(α,β, )
n=
n(1 + n+α)(1 + n+β)(2 + n+α+β)
(1 + 2n+α+β)(2 + 2n+α+β)2(3 + 2n+α+β) (2 + 2n+κ− ,β) (2 + 2n+κ ,β),
and he en ies o B(α,β, )
na e
B(α,β, )
n11 =−n(α+n) −κ− ,−β
(α+β+ 2n+ 2) + (n+ 1)(α+n+ 1) −κ− ,−β
(α+β+ 2n+ 4) ,
B(α,β, )
n21 =κ ,−β(κ− ,β + 2)
(κ− ,β + 2n+ 2) (κ− ,β + 2n+ 4),
B(α,β, )
n12 =−κ− ,−β(κ ,β + 2)
(κ ,β + 2n+ 2) (κ ,β + 2n+ 4),
B(α,β, )
n22 =−n(α+n) +κ ,−β
(α+β+ 2n+ 2) + (n+ 1)(α+n+ 1) +κ ,−β
(α+β+ 2n+ 4) .
P oo . By de ini ion o he polynomials (P(α,β, ,k)
n(x))n≥kin (2.15) one has
(2.27) P(α,β, ,k+1)
n(x) = 1
n−kP(α,β, ,k)
n(x)0.
8 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
The o hogonal monic polynomials P(α,β, ,k)
nn≥ksa is y he h ee e m ecu-
ence ela ion [1, P oposi ion 5.9]
(2.28) xP(α,β, ,k)
n(x) = P(α,β, ,k)
n+1 ( ) + B(k)
nP(α,β, ,k)
n( ) + A(k)
nP(α,β, ,k)
n−1( ),
wi h
(2.29) B(k)
n=B(α+k,β+k, )
n−k, A(k)
n=A(α+k,β+k, )
n−k, n ≥k.
Taking de i a i e in (2.28) i ollows ha
(2.30)
P(α,β, ,k)
n(x) = P(α,β, ,k)
n+1 (x)0+B(k)
nP(α,β, ,k)
n(x)0−xP(α,β, ,k)
n(x)0+A(k)
nP(α,β, ,k)
n−1(x)0.
Replacing kby k+ 1 in (2.28) and using (2.27) one ob ains
x
n−kP(α,β, ,k)
n(x)0= P(α,β, ,k)
n+1 (x)
n+ 1 −k!0
+B(k+1)
n P(α,β, ,k)
n(x)
n−k!0
+A(k+1)
n P(α,β, ,k)
n−1(x)
n−1−k!0
.
By combining (2.31) and (2.30) and using he iden i y
A(α+k,β+k, )
n−k−n−k
n−1−kA(α+k+1,β+k+1, )
n−1−k=1
α+β+k+n+ 2A(α+k,β+k, )
n−k,
he ela ion in P oposi ion 2.1 ollows.

3. The ime-and-band-limi ing p oblem
We s a wi h a e y gene al se up o he ime-and-band limi ing p oblem
which will be applied he e o he example in oduced abo e in Subsec ion 2.2.
Le W=W(x) be a weigh ma ix o size R×Rin he in e al [a, b].
Le Qn(x), n = 0,1,2, ..., be a sequence o eal alued ma ix o hono mal
polynomials wi h espec o he weigh W(x). Conside he ollowing wo Hilbe
spaces: The space L2(W) = L2([a, b], W(x)dx) o all ma ix- alued measu able
unc ions (x), x∈[a, b], sa is ying Rb
a ( (x)W(x) ∗(x)) dx < ∞and he space
`2(MR,N0) o all eal alued R×Rma ix sequences (Cn)n∈N0such ha P∞
n=0 (CnC∗
n)<
∞.
The map F:`2(MR,N0)−→ L2(W) gi en by
(Cn)∞
n=0 7−→ ∞
X
n=0
CnQn(x)
is an isome y. I he polynomials a e dense in L2(W), his map is uni a y wi h he
in e se F−1:L2(W)−→ `2(MR,N0) gi en by
7−→ Cn=Zb
a
(x)W(x)Q∗
n(x)dx.
We deno e ou map by F o emind he eade o he usual Fou ie ans o m.
He e N0 akes up he ole o “ equency space” and he in e al [a, b] he ole o
“physical space”.
The band limi ing ope a o , a le el Nac s on `2(MR,N0) by simply se ing
equal o ze o all he componen s wi h index la ge han N. We deno e i by χN.
The ime limi ing ope a o , a le el Ω, ac s on L2(W) by mul iplica ion by he
TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 15
5. The commu ing h ee-diagonal banded ma ix
Now we conside he case when he global ope a o is gi en by a ull ma ix M
and one ies o look o a idiagonal-banded ma ix Lcommu ing wi h M.
Following Sec ion 3, o each alue o he ime and band limi ing pa ame e s
Nand Ω, one conside s he disc e e ope a o E∗E, ac ing in `2(M2,N0) by a ini e
dimensional block-ma ix M, whe e each block is gi en by he unca ed inne
p oduc s
(5.1) (M)m,n = (E∗E)m,n =ZΩ
0
Qm(x)W(x)Q∗
n(x)dx, 0≤m, n ≤N.
F om [26, Co olla y 3.8] we ha e a cons uc i e way o ob ain he commu ing
idiagonal symme ic ma ix Lusing he commu ing di e en ial ope a o Tin
(4.3) and (4.4).
Indeed, acco ding o [26, P oposi ion 3.4], he e exis ma ices Xnand Ynsuch
ha he ollowing di e en ia ion o mula, in ol ing he sequence o o hono mal
polynomials Qnand he di e en ial ope a o T, holds:
QnT=XnQn+1 +YnQn+X∗
n−1Qn−1,
whe e Xn=hQn+1T, Qni,Yn=hQnT, Qniand assuming Q−1(x) = 0. This is a
consequence o he ac ha QnTis a polynomial o deg ee n+1 and he symme y
o he ope a o T wi h espec o he inne p oduc de ined by W.
In his case, one has ha he idiagonal block-ma ix o size 2(N+1)×2(N+1)
(5.2) L=LΩ,N =









Y0X∗
00 0 0 ···
X0Y1X∗
10 0 ···
0X1Y2X∗
20···
0 0 X2Y3X∗
3···
.
.
..
.
..
.
.......X∗
N−1
0 0 0 ··· XN−1YN









,
commu es wi h he ma ix Min (5.1).
Speci ically, i one conside s he sequence (Pn)n≥0o monic polynomials and
he co esponding ma ix- alued eigen alues Λn(see (2.6)), one has he ollowing
exp ession o he en ies o L(see [26, Co ola y 3.5])
Xn=kPnk−1(Λn+1 + Λn−ΛN+1 −ΛN)kPn+1k,(5.3)
Yn=kPnk−1hPnT, PnikPnk−1.(5.4)
In pa icula , i holds ha XN= 0.
P oposi ion 5.1.Conside he sequence o monic ma ix- alued polynomials
P(k)
n=P(α,β, ,k)
n,n≥k, o hogonal wi h espec o he weigh ma ix W(k)(x)
in (2.14), and i s sequence o eigen alues Λ(k)
nin (2.17), co esponding o he
di e en ial ope a o s D(k)in (2.16). Then, o n∈ {0,1, . . . , N}, he exp essions

16 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
o he en ies o he idiagonal block-ma ix Labo e, a e gi en by
Xn=γnkP(α,β, ,k)
nk−1kP(α,β, ,k)
n+1 k,
Yn=kP(k)
nk−1B(k)
nΛ(k)
n+ Λ(k)
nB(k)
n−2ΩΛ(k)
n+M(k)kP(k)
nk
−DxP(k)
nΛ(k)
N+1 + Λ(k)
N, P(k)
nEkP(k)
nk−1,
whe e
(5.5) γn=−2(n−N)(4 + α+β+n+N),
B(k)
n=B(α+k,β+k, )
n−ka e he coe icien s o he ecu ence ela ion in (2.28) and
M(k)=MD(k)is gi en in (4.5). This ma ix commu es wi h he ma ix Min (5.1)
ob ained by o ming he unca ed inne p oduc s o he o hono mal polynomials
Q(k)
n=kP(k)
nk−1P(k)
n.
P oo . The exp ession o he en ies Xn, ollows di ec ly om (5.3) aking
in o accoun he iden i y
Λ(k)
n+1 + Λ(k)
n−Λ(k)
N+1 −Λ(k)
N=γnI.
Fo he exp ession o he coe icien s Yn, using he ope a o T=TD(k)in (5.4)
we w i e
Yn=kP(k)
nk−1DP(k)
nTD(k), P(k)
nEkP(k)
nk−1,
subs i u ing he exp ession o TD(k)in (4.6) and conside ing he ecu ence ela ion
(2.28) one ob ains
DP(k)
nTD(k), P(k)
nE=DP(k)
nxD(k)+D(k)x−2ΩD(k)−Λ(k)
N+1 + Λ(k)
Nx+M(k), P(k)
ni
=Λ(k)
n+1P(k)
n+1 +B(k)
nΛ(k)
nP(k)
n+A(k)
nΛ(k)
n−1P(k)
n−1+ Λ(k)
nP(k)
n+1 +B(k)
nP(k)
n+A(k)
nP(k)
n−1
−2ΩΛ(k)
nP(k)
n−xP(k)
nΛ(k)
N+1 + Λ(k)
N+P(k)
nM(k), P(k)
n.
Thus, he exp ession o Yn ollows om he o hogonali y o he polynomials P(k)
n.

Fo he case k= 0 one has o he polynomials P(α,β, )
n he explici exp ession
o he squa ed no m P(α,β, )
n
2(see [1, P oposi ion 4.1])
(5.6)
n! B (α+n+ 2, β +n+ 2)
(α+n+ 3 + β)n



(κ ,β + 2) (κ− ,β + 2n+ 4)
κ ,−β(κ ,β + 2n+ 2) 0
0−(κ− ,β + 2) (κ ,β + 2n+ 4)
κ− ,−β(κ− ,β + 2n+ 2)


,
whe e (a)n=a(a+ 1) . . . (a+n−1) deno es he usual Pochhamme symbol and
B(x, y) = R1
0 x−1(1 − )y−1d is he Be a unc ion. Thus, one has he ollowing
co olla y.
TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 17
Co olla y 5.2.Fo he case k= 0, one ob ains o he en ies Xno he
commu ing banded ma ix Lin (5.2), gi en in he p e ious lemma, he explici
exp ession
γnp(κ− ,β + 2n+ 6)(κ ,β + 2n+ 2) 0
0p(κ ,β + 2n+ 6)(κ− ,β + 2n+ 2 ,
whe e γn2=γnΓ(n+2)B(α+n+3,β+n+3)(α+β+n+3)n
n!B(α+n+2,β+n+2)(κ ,β +2n+4)(α+β+n+4)n+1(κ− ,β +2n+4) , wi h γngi en
abo e in (5.5).
5.1. The ad an ages o ha ing a commu ing local ma ix. In he p e ious
sec ion we ha e cons uc ed he ma ix L=LΩ,N , a na ow banded one ha
commu es wi h he ma ix M=MΩ,N , ob ained by o ming he unca ed inne
p oduc s o he no malized ma ix- alued o hogonal polynomials o e a es ic ed
ange in physical space, and has simple spec um. F om a nume ical poin o iew
he en i e pu pose o he sea ch o his second ma ix is ha i educes he p oblem
o compu ing he eigen ec o s o M, a se iously ill-condi ioned p oblem, in o a e y
well condi ioned one. The eigen ec o s o MΩ,N a e impo an since hey gi e he
singula ec o s o he ime-and-band limi ing p oblem desc ibed in Sec ion 3.
Following he ideas o [4, Sec ion 6] and [7, Sec ion 7], we display below he
esul s o some small size nume ical compu a ions ha illus a e he p oblem o
compu ing he eigen ec o s o a ull ma ix, such as MΩ,N , whe e mos o he
eigen alues a e e y close oge he . In each case, we gi e he eigen alues o bo h
he ull ma ix MΩ,N and hose o he na ow banded ma ix LΩ,N .
We use he QR algo i hm as implemen ed in LAPACK. We will deno e by XL
he ma ix o eigen ec o s o LΩ,N (no malized and gi en as columns o XL). We
will deno e by YM he ma ix o eigen ec o s o MΩ,N (no malized and gi en as
columns o YM).
In heo y, he eigen ec o s o MΩ,N should ag ee (up o o de and signs) wi h
hose o LΩ,N . I we compu e he ma ix o inne p oduc s gi en by XT
LYM(whe e
XT
Ldeno es he ansposed ma ix) we expec o ha e he iden i y ma ix up o some
pe mu a ion and possibly some signs due o he no maliza ion o he eigen ec o s
which a e he columns o XLand YM espec i ely.
We choose he pa ame e s α= 5, β= 6, = 2, N= 3 and Ω = 1
50.
In his case, he commu ing ma ix Lo size 8 ×8, has he explici o m:




















3079
50
437√3
221
454q14
221 0 0 0 0 0
437√3
221
4
244
5018√2926
221
50 0 0 0
54q14
221 02359
50
23√627
221
42√38 0 0 0
018√2926
221
5
23√627
221
4
3218
85 02√10374
17 0 0
0 0 2√38 0 1543
50
23√209
119
4
80
√119 0
0 0 0 2√10374
17
23√209
119
4
193036
8075 080√345
119
19
0 0 0 0 80
√119 0631
50
√72105
119
4
0 0 0 0 0 80√345
119
19
√72105
119
4
3494
475




















18 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
The eigen alues o MΩ,N a e
{0.0000445505,7.36172 ×10−8,9.30951 ×10−10,3.26117 ×10−13,8.1543 ×10−15,
8.7946 ×10−19,2.05733 ×10−20,−4.32191 ×10−24}
and hose o LΩ,N a e
{78.0055,52.4488,52.0733,30.9727,30.439,13.7204,12.8858,−0.385373}.
We show he las i e columns o he ma ix XT
LYM, aking he moduli o i s
en ies:












2.48806 ×10−10 2.67928 ×10−90.0036 0.31434 0.9493
1.74104 ×10−93.22846 ×10−80.99999 0.00012 0.00383
8.10627 ×10−10 4.30982 ×10−90.00132 0.94931 0.31434
1.−2.22639 ×10−91.7412 ×10−98.47539 ×10−10 1.19576 ×10−11
2.22637 ×10−91.3.22804 ×10−84.92974 ×10−91.31226 ×10−9
6.77269 ×10−15 2.50177 ×10−14 3.74364 ×10−15 9.49349 ×10−15 1.44166 ×10−15
4.9679 ×10−15 4.77166 ×10−13 3.30976 ×10−14 2.3006 ×10−13 8.94095 ×10−14
2.32541 ×10−16 2.2674 ×10−16 1.34778 ×10−19 2.26045 ×10−17 2.71589 ×10−18












In summa y, obse e ha some o he en ies o XT
LYM, which should be a
pe mu a ion ma ix, a e indeed e y close o he heo e ically co ec alues, while
o he s a e a om hem. The eason is ha mos o he eigen alues o he ull
ma ix o inne p oduc s MΩ,N a e jus oo close oge he . This p oduces nume ical
ins abili y in he compu a ion o he co esponding eigen ec o s. On he o he
hand, all he eigen alues o he commu ing ma ix LΩ,N a e nicely sepa a ed and
in his case, he compu a ion o he co esponding eigen ec o s is a nume ically
e y well condi ioned p oblem.
Re e ences
1. C. Calde ´on and M. M. Cas o, S uc u al o mulas o ma ix- alued o hogonal polynomials
ela ed o 2×2hype geome ic ope a o s, Bull Malays. Sci. Soc. (2022), no. 2, 697–726.
2. C. Calde ´on, Y. Gonz´alez, I. Pacha oni, S. Simondi, and I. Zu i´an, 2 ×2hype geome ic
ope a o s wi h diagonal eigen alues, J. App ox. Theo y, 248:105299, 17pp (2019).
3. W. R. Caspe , F. A. G ¨unbaum, M. Yakimo , I. Zu ian, Ma ix alued disc e e-con inuous
unc ions wi h he p ola e sphe oidal p ope y and bispec ali y (2023), a Xi :2302.05750.
4. M. Cas o and F.A. G ¨unbaum, The Da boux p ocess and ime-and-band limi ing o ma ix
o hogonal polynomials, Linea Algeb a and Applica ions 487 (2015), 328–341.
5. M. Cas o and F. A. G ¨unbaum, Time-and-band limi ing o ma ix o hogonal polynomials
o Jacobi ype, Random Ma ices: Theo y and Applica ions, 06(04):1740001, 12pp (2017).
6. M. Cas o, F.A. G ¨unbaum, I. Pacha oni and I. Zu i´an, A u he look a ime-and-band
limi ing o ma ix o hogonal polynomials, F on ie s in O hogonal Polynomials and q-Se ies,
139–153. Con emp. Ma h. Appl. Monog . Expo. Lec . No es, 1, Wo ld Scien i ic 2018.
7. M. Cas o and F. A. G ¨unbaum, A new p ope y o excep ional o hogonal polynomials (2023),
a Xi :2210.13928.
8. M. J. Can e o, L. Mo al and L. Vel´azquez, Ma ix o hogonal polynomials whose de i a i es
a e also o hogonal, J. App ox. Theo y, 146(2):174–211 (2007).
9. A. J. Du ´an, Ma ix inne p oduc ha ing a ma ix symme ic second o de di e en ial
ope a o , Rocky Moun ain J. Ma h. 27 (1997), 585–600.
10. A. J. Du ´an and P. L´opez, O hogonal ma ix polynomials, La edo Lec u es on O hogonal
Polynomials and Special Func ions, 13–44, Ad . Theo y Spec. Func . O hogonal
Polynomials, No a Sci. Publ., Hauppauge, NY, 2004.
TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 19
11. J. J. Duis e maa and F. A. G ¨unbaum, Di e en ial equa ions in he spec al pa ame e ,
Comm. Ma h. Phys. 103 (1986), 177–240.
12. A. J. Du ´an and F. A. G ¨unbaum, O hogonal ma ix polynomials sa is ying second o de
di e en ial equa ions, In e na ional Ma h. Resea ch No ices 10 (2004), 461–484.
13. A. J. Du ´an and F. A. G ¨unbaum, A su ey on o hogonal polynomials sa is ying second
o de di e en ial equa ions J. Compu . and Applied Ma h. 178 (2005), 169–190.
14. A. J. Du ´an and F. A. G ¨unbaum, S uc u al o mulas o ma ix o hogonal polynomials
sa is ying second o de di e en ial equa ions I, Cons . App ox. 22 (2005), 255–271.
15. A. J. Du ´an and P. L´opez, O hogonal ma ix polynomials, La edo Lec u es on O hogonal
Polynomials and Special Func ions, 13–44, Ad . Theo y Spec. Func . O hogonal
Polynomials, No a Sci. Publ., Hauppauge, NY, 2004.
16. F. A. G ¨unbaum, A new p ope y o ep oducing ke nels o classical o hogonal polynomials,
J. Ma h. Anal. Applic. 95 (1983), 491–500.
17. F. A. G ¨unbaum, Time-band limi ing and he bispec al p oblem, Comm. Pu e Appl. Ma h.
47 (1994), 307–328.
18. F. A. G ¨unbaum, Band- ime-band limi ing in eg al ope a o s and commu ing di e en ial
ope a o s, Algeb a i Analiz 8(1996), 122–126.
19. F. A. G ¨unbaum, Some bispec al musings, CRM P oc. and Lec u e no es ol 14 (1998).
20. F. A. G ¨unbaum, Ma ix alued Jacobi polynomials, Bull. Sciences Ma h 127, n . 3 (2003),
207–214.
21. F. A. G ¨unbaum, L. Longhi and M. Pe ls ad , Di e en ial ope a o s commu ing wi h ini e
con olu ion in eg al ope a o s: some nonabelian examples, SIAM J. Appl. Ma h. 42 (1982),
941–955.
22. F. A. G ¨unbaum, I. Pacha oni and J. A. Ti ao, Ma ix alued sphe ical unc ions associa ed
o he complex p ojec i e plane, J. Func ional Analysis 188 (2002), 350–441.
23. F. A. G ¨unbaum F. A., I. Pacha oni and J. A. Ti ao, Ma ix alued o hogonal polynomials
o he Jacobi ype, Indag. Ma hem. 14, n s. 3,4 (2003), 353 – 366.
24. F. A. G ¨unbaum, I. Pacha oni and I. Zu i´an, Time and band limi ing o ma ix alued
unc ions, an example, SIGMA 11 (2015), 044, 14 pages.
25. F. A. G ¨unbaum, I. Pacha oni and I. Zu i´an, Time and band limi ing o ma ix alued
unc ions: an in eg al and a commu ing di e en ial ope a o , In e se P oblems 33, No. 2
(2017), 025005.
26. F. A. G ¨unbaum, I. Pacha oni, and I. Zu i´an, Bispec ali y and ime-band-limi ing: Ma ix-
alued polynomials, In e na ional Ma h. Resea ch No ices 13, 4016-4036 (2020).
27. E. Koelink, A. de los R´ıos, and P. Rom´an, Ma ix- alued Gegenbaue - ype polynomials,
Cons . App ox., 46(3):459–487 (2017).
28. M. G. K ein, In ini e J-ma ices and a ma ix momen p oblem, Dokl. Akad. Nauk SSSR
69, n . 2 (1949), 125–128.
29. M. G. K ein, Fundamen al aspec s o he ep esen a ion heo y o he mi ian ope a o s wi h
de iciency index (m, m), AMS T ansla ions, Se ies 2, 97, P o idence, Rhode Island (1971),
75–143.
30. H. Landau and H. Pollak, P ola e sphe oidal wa e unc ions, Fou ie Analysis and
Unce ain y, II, Bell Sys em Tech. Jou nal 40, No. 1 (1961), 65–84.
31. H. Landau and H. Pollak, P ola e sphe oidal wa e unc ions, Fou ie Analysis and
Unce ain y, III, Bell Sys em Tech. Jou nal 41, No. 4 (1962), 1295–1336.
32. F. Ma cell´an, A. B anquinho, J. Pe onilho, Classical o hogonal polynomials: a unc ional
app oach, Ac a Appl. Ma h. 34 (3) (1994) 283-303.
33. M.L. Meh a, Random Ma ices, second edi ion, Academic P ess 1991.
34. A. Osipo , V. Rokhlin and H. Xiao, P ola e Sphe oidal Wa e Func ions o O de Ze o,
Ma hema ical Tools o Bandlimi ed App oxima ion, Sp inge 2014.
35. I. Pacha oni and P. Rom´an, A sequence o ma ix alued o hogonal polynomials associa ed
o sphe ical unc ions, Cons uc i e App oxima ion 28, 2 (2008), 127–147.
36. I. Pacha oni and J.A. Ti ao, Ma ix alued o hogonal polynomials a ising om he complex
p ojec i e space Cons uc i e App oxima ion 25, 2 (2007), 177–192.
37. I. Pacha oni and I. Zu i´an, Ma ix Gegenbaue Polynomials: The 2×2Fundamen al Cases,
Cons . App ox., 43(2):253–271 (2016).
38. C. Shannon, A ma hema ical heo y o communica ion, Bell Tech. J. ol 27, 1948, 379–423
(July) and 623–656 (Oc ).
20 M.M. CASTRO, A. FOULQUI´
E-MORENO, AND A. FRADI
39. D. Slepian, P ola e sphe oidal wa e unc ions, Fou ie Analysis and Unce ain y, IV, Bell
Sys em Tech. Jou nal 43, No. 6 (1964), 3009–3058.
40. D. Slepian, On bandwid h, P oc. o IEEE 64, no. 3, Ma ch 1976.
41. D. Slepian, P ola e sphe oidal wa e unc ions, Fou ie Analysis and Unce ain y, V, Bell
Sys em Tech. Jou nal 57, No. 5 (1978), 1371–1430.
42. D. Slepian, Some commen s on Fou ie analysis, Unce ain y and Modeling, SIAM Re iew
25, 3, July 1983, 379–393.
43. D. Slepian and H. Pollak, P ola e sphe oidal wa e unc ions, Fou ie Analysis and
Unce ain y, I, Bell Sys em Tech. Jou nal 40, No. 1 (1961), 43–64.
44. J. A. Ti ao, The ma ix alued hype geome ic equa ion, P oc. Na . Acad. Sci. U.S.A. 100,
n . 14 (2003), 8138–8141.
45. C. A. T acy and H. Widom, Le el spacing dis ibu ion and he Ai y ke nel, Comm. Ma h.
Phys. 159 (1994), 151–174.
46. C. A. T acy and H. Widom, Le el spacing dis ibu ion and he Bessel ke nel, Comm. Ma h.
Phys. 161 (1994), 289–309.
(M.M. Cas o) Depa amen o de Ma em´
a ica Aplicada II and IMUS, Uni e sidad de
Se illa, Escuela Poli ´
ecnica Supe io , c/ Vi gen de A ica 7, 41011, Se illa, Spain
Email add ess, M. Cas o: [email p o ec ed]
(A. Foulqui´e-Mo eno) CIDMA, Depa amen o de Ma em´
a ica, Uni e sidade de A ei o,
Campus de San iago, 3810-193 A ei o, Po ugal
Email add ess, A. Foulqui´e: [email p o ec ed]
(A. F adi) CIDMA, Depa amen o de Ma em´
a ica, Uni e sidade de A ei o, 3810-193
A ei o, Po ugal; Ma hema ical Physics, Special Func ions and Applica ions Labo a o y,
Depa men o Ma hema ics, Ma hema ical Physics, Special Func ions and Applica ions
Labo a o y, The Highe School o Sciences and Technology o Hammam Sousse, Uni e si y
o Sousse, Sousse 4002, Tunisia
Email add ess, A. F adi: [email p o ec ed]