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

PERMISSION TO ARCHIVE PUBLISHED CONTENT Dear, We inform to you that idUS is the Research Repository of the Universidad de Sevilla, which lists the products of research activities of this institution following the Institutional Declaration to promote Open Access. We contact to you in order to ask for permission to archive in idUS under licence CC (by-nc-nd), the following work realized by a member of the Universidad de Sevilla and published with you. A. Author Name, surname: Mirta María Castro Smirnova Email: [email protected] Phone: B. Works to be deposited Title: Time-and-band limiting for matrix-valued orthogonal polynomials related with 2 × 2 hypergeometric operators DOI, ISBN...: https://doi.org/10.1090/conm/807/16164 SOURCE, URL: Contemporary Mathematics, Volume 807, 2024 C. Editor’s authorization to archive in idUS Place and date of sign Place and date of sign Seville, January 22th, 2025 Providence, January 22, 2025 Author’ sign On behalf of Editorial Sign: Name and position Bridget Manzella, Associate Editor for Proceedings Time-and-band limiting for matrix-valued orthogonal polynomials related with 2×2hypergeometric operators M.M. Castro, A. Foulqui´e-Moreno, and A. Fradi Abstract. We consider a family of matrix-valued orthogonal polynomials of size 2 ×2, which are common eigenfunctions of a differential operator of hypergeometric type, in connection with the problem of time-and-band limiting. The problem in question is as follows: given a global operator, defined by an integral kernel or a full matrix, one looks for a local operator given by a second order differential operator or a block tridiagonal matrix respectively, commuting with the global one. We apply the techniques for the construction of the commuting local operator introduced in a previous work by A. Gr¨unbaum, I. Pacharoni and I. Zurrian in the matrix-valued setting, where the role of bispectrality is crucial. This constitutes an illustrative example of an extension to a situation involving matrix orthogonality, of a scalar result that originates in a series of papers by D. Slepian, H. Landau and H. Pollak at Bell Labs in the sixties. 1. Introduction The time-and-band-limiting problem has its origins in the work of D. Slepian, H. Landau and H. Pollak at the Bell Labs back in the 1960’s, [30, 31, 39, 40, 41, 42, 43]. This series of papers was motivated by a question posed by C. Shannon in [38], of what is the best use one can do of the values of the Fourier transform Ff(k) of f, for values of kin the band [−W,W] when f(t) is a time-limited signal with finite support [−T ,T]. The numerically stable reconstruction of ffrom this noisy data leads to the study of an integral operator in L2(−T ,T), given by means of the integral kernel k(t, s) = sin W(t−s)/(t−s). One arrives to the problem of computing numerically most of the eigenfunctions of this operator, which is a very ill-conditioned one, since many of the eigenvalues are very close together. In this sense, D. Slepian, 2020 Mathematics Subject Classification. 33C45, 22E45, 33C47. Key words and phrases. Time-band limiting, Matrix valued orthogonal polynomials. The research of the first author was partially supported by PID2021-124332NB-C21 (FEDER(EU) / Ministerio de Ciencia e Innovaci´on-Agencia Estatal de Investigaci´on) and FQM262 (Junta de Andaluc´ıa). The research of the second and third author was supported by Center of Research and Development in Mathematics and Applications (CIDMA) from University of Aveiro through projects DOI: 10.54499/UIDB/04106/2020 and DOI: 10.54499/UIDP/04106/2020. The third author would also like to express his sincere gratitude to the Mathematical Physics Laboratory, Special Functions and Applications (MaPSFA) for their financial support. 1 2 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI H. Landau and H. Pollak found that an appropriate selfadjoint extension of the operator (e Df)(x) = ((T2−x2)f0(x))0−W2x2f commutes with the given integral operator and has simple spectrum. Thus, the eigenfunctions of both operators are the same. In certain areas of Mathematics, as in this problem, one arrives at a global operator, given by an integral kernel or a full matrix. In certain exceptional cases one can exhibit a differential operator of low order or a matrix with a small number of diagonals (a local operator) which has the same eigenfunctions as the global one. Thus, the numerical computation of the eigenfunctions of the local operator, is numerically stable and much easier than computing numerically the eigenfunctions of the initial global operator in an accurate way (see [34] for more details on computational issues). In this paper we are considering a noncommutative extension of the timeand-band limiting problem described above. We display one illustrative example, involving matrix orthogonality, where this property holds. This question was already taken up in the matrix-valued setting in a few references such as [3, 4, 5, 6, 24, 25, 26], and also for the scalar case in [7, 16, 17, 18, 19, 21]. This problem has also important connections with Random Matrix Theory as one can see in [33, 45, 46]. Specifically, we deal with a family of matrix-valued polynomials (P(α,β,v) n)n≥0 of size 2 ×2, introduced in [2], orthogonal with respect to the weight matrix W(α,β,v)given in (2.7) and (2.8). These polynomials are common eigenfunctions of a differential operator of hypergeometric type (in the sense defined by A. Tirao in [44]), with diagonal matrix eigenvalues, with no repetition in their entries. This family constitutes a new interesting example, among several others in the recent literature, of matrix-valued orthogonal polynomials satisfying second order differential equations, or in other words, exhibiting a bispectral situation according to [11]. The subject of matrix orthogonality, started by Krein in [28] and [29], has received a lot of attention in the last two decades in connection with differential equations after the seminal work of A. Dur´an in [9] (see for instance [12, 13, 14, 20, 22, 23, 27, 35, 36, 37]). The role of bispectrality appears to be crucial in the construction of the timeand-band limiting commuting operator, see [16] for classical orthogonal polynomials and [26] for matrix-valued ones. We apply the techniques in [26] for the construction of the commuting local operators for the case we are considering here. The contents of the present paper are as follows. In the next section, we begin with a general overview of matrix orthogonality and subsequently we introduce the family of polynomials (P(α,β,v) n)n≥0. In particular, in [1] it was shown that the sequence of derivatives of k-th order, k≥1, of the polynomials (P(α,β,v) n)n≥0is also orthogonal, a fact that does not always hold true in the matrix-valued framework (see [14, Section 7]). In this case one has a Pearson type equation (see [1, Theorem 5.4] and (2.18) below), which implies the orthogonality of the derivatives. As it was shown in [8], analogously to the scalar situation, for a given family of matrix-valued orthogonal polynomials (Pn)n≥0, the orthogonality of the derivatives TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 3 is also equivalent to a linear relation between the polynomials Pn,P0 n,P0 n−1and P0 n+1 (see [32]). We show this property for the family of polynomials (P(α,β,v) n)n≥0 considered here, as one may see in Subsection 2.3. In Section 3 we give a general setup for the time-and-band limiting problem in the matrix-valued framework. Therefore, in Section 4, we consider the family of polynomials (P(α,β,v) n)n≥0and its sequence of derivatives (P(α,β,v,k) n)n≥kin connection with this problem. Our main tool for the construction of the commuting differential operator is the Property M hypothesis (see (4.1) below) raised and exploited in [26]. In Section 5, we consider the case when the global operator is given by a full matrix, and construct a commuting tridiagonal banded matrix for the example at hand. Finally, in the line of [4, Section 6] and [7, Section 7], we display the results of some numerical computations, showing the advantages of having such a commuting local matrix. 2. Preliminaries 2.1. Matrix-valued orthogonal polynomials. We start by giving some background on matrix-valued orthogonal polynomials (see [15] for further details). A weight matrix Wis a complex R×Rmatrix-valued integrable function on the interval (a, b), such that Wis positive definite almost everywhere and with finite moments of all orders, i.e., Rb axndW(x)∈CR×R, n ∈N. The weight matrix W induces a Hermitian sesquilinear form, hP, QiW=Zb a P(x)W(x)Q∗(x)dx, for any pair of R×Rmatrix-valued functions P(x) and Q(x), where Q∗(x) denotes the conjugate transpose of Q(x). We consider a sequence (Pn)n≥0of orthogonal polynomials with respect to a weight matrix W, where Pn(x) = xnCn+xn−1Cn−1+···+C0, Ci∈CR×R, i = 0,1, . . . , n, is a matrix-valued polynomial with non-singular leading coefficient Cn, and hPn, PmiW = ∆nδn,m, where ∆n,n≥0, is a positive definite matrix. When ∆n=I, here I denotes the identity matrix, we say that the polynomials (Pn)n≥0are orthonormal. In particular, when the leading coefficient of Pn(x), n≥0, is the identity matrix, we say that the polynomials (Pn)n≥0are monic. Given a weight matrix W, there exists a unique sequence of monic orthogonal polynomials (Pn)n≥0in CR×R[x], any other sequence of orthogonal polynomials (Qn)n≥0can be written as Qn(x) = KnPn(x) for some non-singular matrix Kn. Any sequence of monic orthogonal matrix-valued polynomials (Pn)n≥0satisfies a three-term recurrence relation (2.1) xPn(x) = Pn+1(x) + BnPn(x) + AnPn−1(x),for n∈N0, where P−1(x) = 0, P0(x) = I. The R×Rmatrix coefficients Anand Bnenjoy certain properties, in particular, Anis non-singular for any n. Let Dbe a right-hand side ordinary differential operator with matrix valued polynomial coefficients, (2.2) D= s X i=0 ∂iFi(x), ∂i=di dxi. 4 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI The operator Dacts on a polynomial function P(x) as P D =Ps i=0 ∂iPFi(x). We say that the differential operator Dis symmetric with respect to Wif (2.3) hPD, QiW=hP, QDiW,for all P, Q ∈CR×R[x]. A necessary and sufficient condition for a matrix-valued differential operator to be symmetric is given in [12, Theorem 3.1]. According to this result, the differential operator D=d2 dx2F2(x) + d dxF1(x) + F0is symmetric with respect to Wif and only if 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 family of matrix-valued orthogonal polynomials (Pn)n≥0satisfying second order differential equations, gives a basis of polynomials in CR×R[x] that are joint eigenfunctions of a fixed differential operator Dof order two, as given in (2.2), with matrix-valued eigenvalue Λn (2.6) DPn(x)=ΛnPn(x). At the same time, they are eigenfunctions of a second order difference operator B, given by an infinite tridiagonal block-matrix whose entries are given by the coefficients of the three-term recurrence relation in (2.1), with eigenvalue θ(x) = x, that is to say, BPn=θ(x)Pn. In this sense, one says that the family (Pn)n≥0is bispectral, in the spirit of [11]. 2.2. The family of matrix-valued orthogonal polynomials. In [2] the authors introduce a Jacobi type weight matrix W(α,β,v)(x) and a differential operator D(α,β,v)such that D(α,β,v)is symmetric with respect to the weight matrix W(α,β,v)(x). Let α,β,v∈R,α, β > −1 and |α−β|<|v|< α +β+ 2. We consider the weight matrix function (2.7) W(α,β,v)(x) = xα(1 −x)βf W(α,β,v)(x),for x∈(0,1), with f W(α,β,v)(x) = (2.8)     v(κv,β + 2) κv,−β x2−(κv,β + 2) x+ (α+ 1) (α+β+ 2)x−(α+ 1) (α+β+ 2)x−(α+ 1) −v(κ−v,β + 2) κ−v,−β x2−(κ−v,β + 2) x+ (α+ 1)    , where, for the sake of clearness in the rest of the paper, we use the notation: (2.9) κ±v,±β=α±v±β . W(α,β,v)is an irreducible weight matrix and the hypergeometric type differential operator given by TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 5 (2.10) D(α,β,v)=d2 dx2F2(x) + d dtF1(x) + F0(x), where (2.11) F2(x) = x(1 −x), F1(x) = C∗−xU and F0(x) = −V, and (2.12) C= α+ 1 −κ−v,−β v κv,−β v −κ−v,−β vα+1+κv,−β v , U = (α+β+ 4) Iand V=v0 0 0, is symmetric with respect to W(α,β,v). As shown in [2, Theorem 4.3], the sequence of matrix-valued monic orthogonal polynomials P(α,β,v) nn≥0associated with the weight function W(α,β,v)(x) are common eigenfunctions of the differential operator D(α,β,v)with diagonal eigenvalues (2.13) Λ(α,β,v) n=λn0 0µn,λn=−n(n−1) −n(α+β+ 4) −v, µn=−n(n−1) −n(α+β+ 4) . Consider now the weight matrix given by (2.14) W(k)(x) = W(α+k,β+k,v)(x) and let dk dxkP(α,β,v) n(x) be the derivative of order kof the monic polynomial P(α,β,v) n(x), for n≥k. Then (2.15) P(α,β,v,k) n(x) = (n−k)! n! dk dxkP(α,β,v) n(x) are monic polynomials of degree n−kfor all n≥k. Observe that for the case k= 0 one has the family of polynomials defined initially by the weight W(α,β,v)(x) in (2.7). Consider the operator (2.16) D(k)=D(α,β,v,k)=d2 dx2x(1 −x) + d dt((C(k))∗−xU(k))−V, with C(k)=C+kI, U(k)=U+ 2kI = (α+β+ 4 + 2k) I, As shown in [1, Section 5], the sequence of derivatives are common eigenfunctions of the hypergeometric type operator D(k)with diagonal matrix eigenvalues (2.17) Λ(k) n= Λ(α,β,v,k)= λ(k) n0 0µ(k) n!,λ(k) n=−(n−k) (α+β+3+n+k)−v, µ(k) n=−(n−k) (α+β+3+n+k). 6 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI Moreover, the authors prove that the sequence of derivatives P(α,β,v,k) n(x) are orthogonal with respect to the weight matrix W(k)(x), which is a consequence of a Pearson type equation: W(k)(x) Φ(k)(x)0=W(k)(x) Ψ(k)(x), k ∈N,with(2.18) Φ(k)(x) = Ak 2x2+Ak 1x+Ak 0and Ψ(k)(x) = Bk 1x+Bk 0, where Ak 2=   −κv,β + 2(k+ 2) κv,β + 2(k+ 1) 0 0−κ−v,β + 2(k+ 2) κ−v,β + 2(k+ 1)    , (2.19) Ak 1=2 (κ−v,β + 2(k+ 1))(κv,β + 2(k+ 1)) 0κv,−β κ−v,−β0−Ak 2, (2.20) Ak 0=κv,−βκ−v,−β v(κ−v,β + 2(k+ 1))(κv,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 v−κ−v,−β0 0κv,−βAk 2 (2.23) +1 2vα+β+ 2k+ 4 vAk 1+Bk 1−κ−v,β −2(k+ 1) 0 0κv,β + 2(k+ 1). Particularly, one has that (2.24) W(k+1) =W(k)Φ(k)(x), where W(k+1) is precisely the orthogonality weight associated to the sequence of derivatives of order k+ 1. This relation allows to construct symmetric differential operators with respect to the weight W(k)using the entries of the equation. Indeed, in this case one has that the differential operator (see [1, Corollary 6.4]) (2.25) E(k)=d2 dx2(Φ(k)(x))∗+d dx(Ψ(k)(x))∗ is symmetric with respect to W(k)(x) for all k∈N0.Moreover, the polynomials P(α,β,v,k) n+kn≥0are eigenfunctions of the operator E(k)with eigenvalue (2.26) ΛnE(k)=n(n+α+β+ 3 + 2k)Ak 2. Thus, for each fixed value of k≥0, we have two different symmetric operators D(k)and E(k)in the algebra of matrix differential operators having as common eigenfunctions the sequence of polynomials P(α,β,v,k) n(x)n≥0, orthogonal with respect TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 7 to the weight W=W(k). This fact will be very useful to construct commuting differential operators in the time-and-band-limiting framework below in section 4. In the next section we complete the characterization of the property of orthogonality of the sequence of derivatives P(α,β,v,k) n(x)n≥kfor the family of matrix-valued polynomials P(α,β,v) n(x) considered here. 2.3. A characterization property for the orthogonality of the derivatives. Following the results in [32] for classical orthogonal polynomials and the result in [8, Theorem 3.14] we show the following recurrence relation involving the polynomials P(α,β,v,k) n(x) and the sequence of their derivatives, which are at the same time orthogonal matrix-valued polynomials with respect to the weight matrix W(k+1) in (2.14). Proposition 2.1.The monic orthogonal polynomials (P(α,β,v,k) n(x))n≥kand their derivatives satisfy the recurrence relation P(α,β,v,k) n(x) = 1 n+ 1 −kP(α,β,v,k) n+1 (x)0 +B(α+k,β+k,v) n−k−B(α+k+1,β+k+1,v) n−1−kP(α,β,v,k) n(x)0 −1 α+β+k+n+ 2A(α+k,β+k,v) n−kP(α,β,v,k) n−1(x)0, where, for n≥0, the expressions of the coefficients A(α,β,v) nare (see [1, Proposition 2.3] and [2, Theorem 3.12]) A(α,β,v) n=a(α,β,v) n(4 + 2n+κ−v,β) (2n+κv,β) 0 0 (4 + 2n+κv,β ) (2n+κ−v,β ), with a(α,β,v) 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−v,β) (2 + 2n+κv,β), and the entries of B(α,β,v) nare B(α,β,v) n11 =−n(α+n)v−κ−v,−β (α+β+ 2n+ 2)v+ (n+ 1)(α+n+ 1)v−κ−v,−β (α+β+ 2n+ 4)v, B(α,β,v) n21 =κv,−β(κ−v,β + 2) v(κ−v,β + 2n+ 2) (κ−v,β + 2n+ 4), B(α,β,v) n12 =−κ−v,−β(κv,β + 2) v(κv,β + 2n+ 2) (κv,β + 2n+ 4), B(α,β,v) n22 =−n(α+n)v+κv,−β (α+β+ 2n+ 2)v+ (n+ 1)(α+n+ 1)v+κv,−β (α+β+ 2n+ 4)v. Proof. By definition of the polynomials (P(α,β,v,k) n(x))n≥kin (2.15) one has (2.27) P(α,β,v,k+1) n(x) = 1 n−kP(α,β,v,k) n(x)0. 8 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI The orthogonal monic polynomials P(α,β,v,k) nn≥ksatisfy the three term recurrence relation [1, Proposition 5.9] (2.28) xP(α,β,v,k) n(x) = P(α,β,v,k) n+1 (t) + B(k) nP(α,β,v,k) n(t) + A(k) nP(α,β,v,k) n−1(t), with (2.29) B(k) n=B(α+k,β+k,v) n−k, A(k) n=A(α+k,β+k,v) n−k, n ≥k. Taking derivative in (2.28) it follows that (2.30) P(α,β,v,k) n(x) = P(α,β,v,k) n+1 (x)0+B(k) nP(α,β,v,k) n(x)0−xP(α,β,v,k) n(x)0+A(k) nP(α,β,v,k) n−1(x)0. Replacing kby k+ 1 in (2.28) and using (2.27) one obtains x n−kP(α,β,v,k) n(x)0= P(α,β,v,k) n+1 (x) n+ 1 −k!0 +B(k+1) n P(α,β,v,k) n(x) n−k!0 +A(k+1) n P(α,β,v,k) n−1(x) n−1−k!0 . By combining (2.31) and (2.30) and using the identity A(α+k,β+k,v) n−k−n−k n−1−kA(α+k+1,β+k+1,v) n−1−k=1 α+β+k+n+ 2A(α+k,β+k,v) n−k, the relation in Proposition 2.1 follows.  3. The time-and-band-limiting problem We start with a very general setup for the time-and-band limiting problem which will be applied here to the example introduced above in Subsection 2.2. Let W=W(x) be a weight matrix of size R×Rin the interval [a, b]. Let Qn(x), n = 0,1,2, ..., be a sequence of real valued matrix orthonormal polynomials with respect to the weight W(x). Consider the following two Hilbert spaces: The space L2(W) = L2([a, b], W(x)dx) of all matrix-valued measurable functions f(x), x∈[a, b], satisfying Rb atr (f(x)W(x)f∗(x)) dx < ∞and the space `2(MR,N0) of all real valued R×Rmatrix sequences (Cn)n∈N0such that P∞ n=0 tr (CnC∗ n)< ∞. The map F:`2(MR,N0)−→ L2(W) given by (Cn)∞ n=0 7−→ ∞ X n=0 CnQn(x) is an isometry. If the polynomials are dense in L2(W), this map is unitary with the inverse F−1:L2(W)−→ `2(MR,N0) given by f7−→ Cn=Zb a f(x)W(x)Q∗ n(x)dx. We denote our map by Fto remind the reader of the usual Fourier transform. Here N0takes up the role of “frequency space” and the interval [a, b] the role of “physical space”. The band limiting operator, at level Nacts on `2(MR,N0) by simply setting equal to zero all the components with index larger than N. We denote it by χN. The time limiting operator, at level Ω, acts on L2(W) by multiplication by the TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 15 5. The commuting three-diagonal banded matrix Now we consider the case when the global operator is given by a full matrix M and one tries to look for a tridiagonal-banded matrix Lcommuting with M. Following Section 3, for each value of the time and band limiting parameters Nand Ω, one considers the discrete operator E∗E, acting in `2(M2,N0) by a finite dimensional block-matrix M, where each block is given by the truncated inner products (5.1) (M)m,n = (E∗E)m,n =ZΩ 0 Qm(x)W(x)Q∗ n(x)dx, 0≤m, n ≤N. From [26, Corollary 3.8] we have a constructive way to obtain the commuting tridiagonal symmetric matrix Lusing the commuting differential operator Tin (4.3) and (4.4). Indeed, according to [26, Proposition 3.4], there exist matrices Xnand Ynsuch that the following differentiation formula, involving the sequence of orthonormal polynomials Qnand the differential operator T, holds: QnT=XnQn+1 +YnQn+X∗ n−1Qn−1, where Xn=hQn+1T, Qni,Yn=hQnT, Qniand assuming Q−1(x) = 0. This is a consequence of the fact that QnTis a polynomial of degree n+1 and the symmetry of the operator T with respect to the inner product defined by W. In this case, one has that the tridiagonal block-matrix of 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          , commutes with the matrix Min (5.1). Specifically, if one considers the sequence (Pn)n≥0of monic polynomials and the corresponding matrix-valued eigenvalues Λn(see (2.6)), one has the following expression for the entries of L(see [26, Corrolary 3.5]) Xn=kPnk−1(Λn+1 + Λn−ΛN+1 −ΛN)kPn+1k,(5.3) Yn=kPnk−1hPnT, PnikPnk−1.(5.4) In particular, it holds that XN= 0. Proposition 5.1.Consider the sequence of monic matrix-valued polynomials P(k) n=P(α,β,v,k) n,n≥k, orthogonal with respect to the weight matrix W(k)(x) in (2.14), and its sequence of eigenvalues Λ(k) nin (2.17), corresponding to the differential operators D(k)in (2.16). Then, for n∈ {0,1, . . . , N}, the expressions 16 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI of the entries of the tridiagonal block-matrix Labove, are given by Xn=γnkP(α,β,v,k) nk−1kP(α,β,v,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, where (5.5) γn=−2(n−N)(4 + α+β+n+N), B(k) n=B(α+k,β+k,v) n−kare the coefficients of the recurrence relation in (2.28) and M(k)=MD(k)is given in (4.5). This matrix commutes with the matrix Min (5.1) obtained by forming the truncated inner products of the orthonormal polynomials Q(k) n=kP(k) nk−1P(k) n. Proof. The expression of the entries Xn, follows directly from (5.3) taking into account the identity Λ(k) n+1 + Λ(k) n−Λ(k) N+1 −Λ(k) N=γnI. For the expression of the coefficients Yn, using the operator T=TD(k)in (5.4) we write Yn=kP(k) nk−1DP(k) nTD(k), P(k) nEkP(k) nk−1, substituting the expression of TD(k)in (4.6) and considering the recurrence relation (2.28) one obtains 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, the expression of Ynfollows from the orthogonality of the polynomials P(k) n.  For the case k= 0 one has for the polynomials P(α,β,v) nthe explicit expression of the squared norm P(α,β,v) n 2(see [1, Proposition 4.1]) (5.6) n!vB (α+n+ 2, β +n+ 2) (α+n+ 3 + β)n    (κv,β + 2) (κ−v,β + 2n+ 4) κv,−β(κv,β + 2n+ 2) 0 0−(κ−v,β + 2) (κv,β + 2n+ 4) κ−v,−β(κ−v,β + 2n+ 2)   , where (a)n=a(a+ 1) . . . (a+n−1) denotes the usual Pochhammer symbol and B(x, y) = R1 0tx−1(1 −t)y−1dt is the Beta function. Thus, one has the following corollary. TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 17 Corollary 5.2.For the case k= 0, one obtains for the entries Xnof the commuting banded matrix Lin (5.2), given in the previous lemma, the explicit expression fγnp(κ−v,β + 2n+ 6)(κv,β + 2n+ 2) 0 0p(κv,β + 2n+ 6)(κ−v,β + 2n+ 2 , where fγn2=γnΓ(n+2)B(α+n+3,β+n+3)(α+β+n+3)n n!B(α+n+2,β+n+2)(κv,β +2n+4)(α+β+n+4)n+1(κ−v,β +2n+4) , with γngiven above in (5.5). 5.1. The advantages of having a commuting local matrix. In the previous section we have constructed the matrix L=LΩ,N , a narrow banded one that commutes with the matrix M=MΩ,N , obtained by forming the truncated inner products of the normalized matrix-valued orthogonal polynomials over a restricted range in physical space, and has simple spectrum. From a numerical point of view the entire purpose of the search for this second matrix is that it reduces the problem of computing the eigenvectors of M, a seriously ill-conditioned problem, into a very well conditioned one. The eigenvectors of MΩ,N are important since they give the singular vectors of the time-and-band limiting problem described in Section 3. Following the ideas of [4, Section 6] and [7, Section 7], we display below the results of some small size numerical computations that illustrate the problem of computing the eigenvectors of a full matrix, such as MΩ,N , where most of the eigenvalues are very close together. In each case, we give the eigenvalues of both the full matrix MΩ,N and those of the narrow banded matrix LΩ,N . We use the QR algorithm as implemented in LAPACK. We will denote by XL the matrix of eigenvectors of LΩ,N (normalized and given as columns of XL). We will denote by YMthe matrix of eigenvectors of MΩ,N (normalized and given as columns of YM). In theory, the eigenvectors of MΩ,N should agree (up to order and signs) with those of LΩ,N . If we compute the matrix of inner products given by XT LYM(where XT Ldenotes the transposed matrix) we expect to have the identity matrix up to some permutation and possibly some signs due to the normalization of the eigenvectors which are the columns of XLand YMrespectively. We choose the parameters α= 5, β= 6, v= 2, N= 3 and Ω = 1 50. In this case, the commuting matrix Lof size 8 ×8, has the explicit form:                     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 eigenvalues of MΩ,N are {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 those of LΩ,N are {78.0055,52.4488,52.0733,30.9727,30.439,13.7204,12.8858,−0.385373}. We show the last five columns of the matrix XT LYM, taking the moduli of its entries:             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 summary, observe that some of the entries of XT LYM, which should be a permutation matrix, are indeed very close to the theoretically correct values, while others are far from them. The reason is that most of the eigenvalues of the full matrix of inner products MΩ,N are just too close together. This produces numerical instability in the computation of the corresponding eigenvectors. On the other hand, all the eigenvalues of the commuting matrix LΩ,N are nicely separated and in this case, the computation of the corresponding eigenvectors is a numerically very well conditioned problem. References 1. C. Calder´on and M. M. Castro, Structural formulas for matrix-valued orthogonal polynomials related to 2×2hypergeometric operators, Bull Malays. Sci. Soc. (2022), no. 2, 697–726. 2. C. Calder´on, Y. Gonz´alez, I. Pacharoni, S. Simondi, and I. Zurri´an, 2 ×2hypergeometric operators with diagonal eigenvalues, J. Approx. Theory, 248:105299, 17pp (2019). 3. W. R. Casper, F. A. Gr¨unbaum, M. Yakimov, I. Zurrian, Matrix valued discrete-continuous functions with the prolate spheroidal property and bispectrality (2023), arXiv:2302.05750. 4. M. Castro and F.A. Gr¨unbaum, The Darboux process and time-and-band limiting for matrix orthogonal polynomials, Linear Algebra and Applications 487 (2015), 328–341. 5. M. Castro and F. A. Gr¨unbaum, Time-and-band limiting for matrix orthogonal polynomials of Jacobi type, Random Matrices: Theory and Applications, 06(04):1740001, 12pp (2017). 6. M. Castro, F.A. Gr¨unbaum, I. Pacharoni and I. Zurri´an, A further look at time-and-band limiting for matrix orthogonal polynomials, Frontiers in Orthogonal Polynomials and q-Series, 139–153. Contemp. Math. Appl. Monogr. Expo. Lect. Notes, 1, World Scientific 2018. 7. M. Castro and F. A. Gr¨unbaum, A new property of exceptional orthogonal polynomials (2023), arXiv:2210.13928. 8. M. J. Cantero, L. Moral and L. Vel´azquez, Matrix orthogonal polynomials whose derivatives are also orthogonal, J. Approx. Theory, 146(2):174–211 (2007). 9. A. J. Dur´an, Matrix inner product having a matrix symmetric second order differential operator, Rocky Mountain J. Math. 27 (1997), 585–600. 10. A. J. Dur´an and P. L´opez, Orthogonal matrix polynomials, Laredo Lectures on Orthogonal Polynomials and Special Functions, 13–44, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004. TIME-AND-BAND LIMITING AND MATRIX ORTHOGONALITY 19 11. J. J. Duistermaat and F. A. Gr¨unbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240. 12. A. J. Dur´an and F. A. Gr¨unbaum, Orthogonal matrix polynomials satisfying second order differential equations, International Math. Research Notices 10 (2004), 461–484. 13. A. J. Dur´an and F. A. Gr¨unbaum, A survey on orthogonal polynomials satisfying second order differential equations J. Comput. and Applied Math. 178 (2005), 169–190. 14. A. J. Dur´an and F. A. Gr¨unbaum, Structural formulas for matrix orthogonal polynomials satisfying second order differential equations I, Constr. Approx. 22 (2005), 255–271. 15. A. J. Dur´an and P. L´opez, Orthogonal matrix polynomials, Laredo Lectures on Orthogonal Polynomials and Special Functions, 13–44, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004. 16. F. A. Gr¨unbaum, A new property of reproducing kernels of classical orthogonal polynomials, J. Math. Anal. Applic. 95 (1983), 491–500. 17. F. A. Gr¨unbaum, Time-band limiting and the bispectral problem, Comm. Pure Appl. Math. 47 (1994), 307–328. 18. F. A. Gr¨unbaum, Band-time-band limiting integral operators and commuting differential operators, Algebra i Analiz 8(1996), 122–126. 19. F. A. Gr¨unbaum, Some bispectral musings, CRM Proc. and Lecture notes vol 14 (1998). 20. F. A. Gr¨unbaum, Matrix valued Jacobi polynomials, Bull. Sciences Math 127, nr. 3 (2003), 207–214. 21. F. A. Gr¨unbaum, L. Longhi and M. Perlstadt, Differential operators commuting with finite convolution integral operators: some nonabelian examples, SIAM J. Appl. Math. 42 (1982), 941–955. 22. F. A. Gr¨unbaum, I. Pacharoni and J. A. Tirao, Matrix valued spherical functions associated to the complex projective plane, J. Functional Analysis 188 (2002), 350–441. 23. F. A. Gr¨unbaum F. A., I. Pacharoni and J. A. Tirao, Matrix valued orthogonal polynomials of the Jacobi type, Indag. Mathem. 14, nrs. 3,4 (2003), 353 – 366. 24. F. A. Gr¨unbaum, I. Pacharoni and I. Zurri´an, Time and band limiting for matrix valued functions, an example, SIGMA 11 (2015), 044, 14 pages. 25. F. A. Gr¨unbaum, I. Pacharoni and I. Zurri´an, Time and band limiting for matrix valued functions: an integral and a commuting differential operator, Inverse Problems 33, No. 2 (2017), 025005. 26. F. A. Gr¨unbaum, I. Pacharoni, and I. Zurri´an, Bispectrality and time-band-limiting: Matrixvalued polynomials, International Math. Research Notices 13, 4016-4036 (2020). 27. E. Koelink, A. de los R´ıos, and P. Rom´an, Matrix-valued Gegenbauer-type polynomials, Constr. Approx., 46(3):459–487 (2017). 28. M. G. Krein, Infinite J-matrices and a matrix moment problem, Dokl. Akad. Nauk SSSR 69, nr. 2 (1949), 125–128. 29. M. G. Krein, Fundamental aspects of the representation theory of hermitian operators with deficiency index (m, m), AMS Translations, Series 2, 97, Providence, Rhode Island (1971), 75–143. 30. H. Landau and H. Pollak, Prolate spheroidal wave functions, Fourier Analysis and Uncertainty, II, Bell System Tech. Journal 40, No. 1 (1961), 65–84. 31. H. Landau and H. Pollak, Prolate spheroidal wave functions, Fourier Analysis and Uncertainty, III, Bell System Tech. Journal 41, No. 4 (1962), 1295–1336. 32. F. Marcell´an, A. Branquinho, J. Petronilho, Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (3) (1994) 283-303. 33. M.L. Mehta, Random Matrices, second edition, Academic Press 1991. 34. A. Osipov, V. Rokhlin and H. Xiao, Prolate Spheroidal Wave Functions of Order Zero, Mathematical Tools for Bandlimited Approximation, Springer 2014. 35. I. Pacharoni and P. Rom´an, A sequence of matrix valued orthogonal polynomials associated to spherical functions, Constructive Approximation 28, 2 (2008), 127–147. 36. I. Pacharoni and J.A. Tirao, Matrix valued orthogonal polynomials arising from the complex projective space Constructive Approximation 25, 2 (2007), 177–192. 37. I. Pacharoni and I. Zurri´an, Matrix Gegenbauer Polynomials: The 2×2Fundamental Cases, Constr. Approx., 43(2):253–271 (2016). 38. C. Shannon, A mathematical theory of communication, Bell Tech. J. vol 27, 1948, 379–423 (July) and 623–656 (Oct). 20 M.M. CASTRO, A. FOULQUI´ E-MORENO, AND A. FRADI 39. D. Slepian, Prolate spheroidal wave functions, Fourier Analysis and Uncertainty, IV, Bell System Tech. Journal 43, No. 6 (1964), 3009–3058. 40. D. Slepian, On bandwidth, Proc. of IEEE 64, no. 3, March 1976. 41. D. Slepian, Prolate spheroidal wave functions, Fourier Analysis and Uncertainty, V, Bell System Tech. Journal 57, No. 5 (1978), 1371–1430. 42. D. Slepian, Some comments on Fourier analysis, Uncertainty and Modeling, SIAM Review 25, 3, July 1983, 379–393. 43. D. Slepian and H. Pollak, Prolate spheroidal wave functions, Fourier Analysis and Uncertainty, I, Bell System Tech. Journal 40, No. 1 (1961), 43–64. 44. J. A. Tirao, The matrix valued hypergeometric equation, Proc. Nat. Acad. Sci. U.S.A. 100, nr. 14 (2003), 8138–8141. 45. C. A. Tracy and H. Widom, Level spacing distribution and the Airy kernel, Comm. Math. Phys. 159 (1994), 151–174. 46. C. A. Tracy and H. Widom, Level spacing distribution and the Bessel kernel, Comm. Math. Phys. 161 (1994), 289–309. (M.M. Castro) Departamento de Matem´ atica Aplicada II and IMUS, Universidad de Sevilla, Escuela Polit´ ecnica Superior, c/ Virgen de Africa 7, 41011, Sevilla, Spain Email address, M. Castro: [email protected] (A. Foulqui´e-Moreno) CIDMA, Departamento de Matem´ atica, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal Email address, A. Foulqui´e: [email protected] (A. Fradi) CIDMA, Departamento de Matem´ atica, Universidade de Aveiro, 3810-193 Aveiro, Portugal; Mathematical Physics, Special Functions and Applications Laboratory, Department of Mathematics, Mathematical Physics, Special Functions and Applications Laboratory, The Higher School of Sciences and Technology of Hammam Sousse, University of Sousse, Sousse 4002, Tunisia Email address, A. Fradi: [email protected]