Time-and-band limiting for matrix-valued orthogonal polynomials related with 2 × 2 hypergeometric operators

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. 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