Time-and-band limiting for matrix orthogonal polynomials of Jacobi type

TIME-AND-BAND LIMITING FOR MATRIX ORTHOGONAL POLYNOMIALS OF JACOBI TYPE M. CASTRO AND F. A. GR¨ UNBAUM Abstract. We extend to a situation involving matrix valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathematics and signal processing. We consider a case of matrix valued Jacobi polynomials which arises from the study of representations of SU(N), a group that plays in important role in Random matrix theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix valued situation. 1. Introduction The issue of finding a differential operator that would commute with a naturally appearing integral one has a long history and plays an important role in Random Matrix theory. One can see its relevance in M. Mehta grounbreaking book [22] in connection with the kernel sin(W(x−y))/(x−y) and in some of the work of C. Tracy and H. Widom such as in [33, 34], in connection with the kernels that come about by replacing the exponentials of the Fourier transform by either Bessel or Airy functions. The same topic has arisen in other contexts, such as in signal processing in the hands of D. Slepian, H. Landau and H. Pollak back in the 1960’s, see [20, 21, 26, 27, 28, 29, 30, 31]. These are not the only areas where this problem raises its head, for a recent treatment of the many numerical advantages that derive from such a remarkable phenomenon see [23]. In this paper, which follows related work in [1, 2, 16, 17] we consider a family of matrix valued polynomials which are orthogonal with respect to a matrix valued weight, a theory started by M.G. Krein, see [18, 19], and have the extra property of being joint eigenfunctions of a second order differential operator with matrix valued coefficients. The particular example that we study, depending on three free parameters, was first considered in [25] in connection with representations of SU(N) and its matrix valued spherical functions. Here one is in the presence of a “bispectral situation” a subject which has been explored in the matrix valued case in several papers. A short list includes [3, 5, 6, 7, 14, 15]. 2010 Mathematics Subject Classification. 33C45, 22E45, 33C47. Key words and phrases. Time-band limiting, Matrix valued orthogonal polynomials. The work of the first author is partially supported by MTM2015-6588-C4-1-P (Ministerio de Econom´ıa y Competividad), FQM-262, FQM-7276 (Junta de Andaluc´ıa), Feder Funds (European Union) and the program Campus de Excelencia Internacional of Ministerio de Educaci´on, Cultura y Deporte. The work of the second author is partially supported by AFOSR, through FA9550-16-1-0175. 1 2 M. CASTRO AND F. A. GR ¨ UNBAUM Although much of this work can be carried out for matrices of arbitrary size we restrict our attention to the 2 ×2 case. One word about “names” for the polynomials in question. Maybe the first paper to give explict examples of matrix valued polynomials of Jacobi type is [12]. The ones in [25] depend on more parameters and are more naturally connected with the group representation theory of SU(N). We give now a “bird’s view” of the contents of this paper. Starting with a nice “essentially hypergeometric” sequence of orthogonal polynomials denoted by Pn(x) with xin the interval [0,1] and a weight matrix depending on three parameters, α, β, k, and given explicitly by Z(x) = (1 −x)αxβk(1 −x)+(β+ 1 −k) (β+ 1 −k)x (β+ 1 −k)x(β+ 1 −k)x2 we consider the problem of “time-and-band limiting”: choose Tin the interval (0,1) and a natural number N. Time limiting will amount to restricting matrix valued functions originally defined in [0,1] to the interval [0, T]. Band limiting will amount to projecting such a function onto the linear span of the first Northogonal polynomials with respect to Z(x). A more detailed description of “time-and-band limiting”, including the way in which an integral operator appears naturally, will be given in the next section. In the simplest scalar valued case, one runs into the famous sinc kernel that appears in the work of M. Mehta and in the signal processing setup alluded to early. This leads to the “prolate spheroidal wave functions” in [22]. Now we start introducing the characters of our story. The sequence of orthogonal polynomials Pn(x), to be constructed later on, satisfy a three term recursion relation with matrix valued coefficients. This is a direct consequence, as noted first by M.G. Krein of their orthogonality with respect to the weight Z(x). They are also eigenfunctions of the differential operator D=d dx2 x(1 −x) + d dx(C−xU)−V with C=β+ 1 1 0β+ 3, U =α+β+ 3 0 0α+β+ 4 V=0 0 k−β−1α+β+ 2 −k This operator, as well as all differential operators that will appear later, acts on the right and we have PnD=∧nPn with ∧n=−n(n+α+β+ 2) 0 0−n(n+α+β+ 3) −(α+β+ 2 −k). TIME-AND-BAND LIMITING FOR MATRIX ORTHOGONAL POLYNOMIALS OF JACOBI TYPE 3 In reference [24] one finds an explicit construction, in terms of the matrix-valued hypergeometric functions introduced in [32], of a family of polynomials with the property above. Our choice of polynomials Pn(x) will be different than that in [24], but the entries of our polynomials have a definite hypergeometric flavour too. We will also introduce a family Qn(x) of polynomials that are orthonormal with respect to the weight above and satisfy the same differential equation. Finally we can mention the main characters of our story: In terms of the Qn(x) above the integral operator will have a kernel given by k(x, y) = N X j=0 Q∗ j(x)Qj(y). and the differential operator commuting with it (in the appropriate interval and with appropriate boundary conditions) will be given by (x−T)D−d dxx(1 −x) +x(N2+ (α+β+ 3)(N+ 1)) α+β+N+ 2 x(k−β−1) x[N2+ (α+β+ 4)N+ 2α+ 2β−k+ 6] + β. We now proceed to a more in depth discussion. 2. A more detailed introduction We start with a very general setup of the time-band limiting problem which will be applied later on to the case at hand. Let Z=Z(x) be a weight matrix of size 2 in the interval [0,1]. By this we mean a complex 2×2-matrix valued integrable function Zon the interval [0,1] such that Z(x) is positive definitive almost everywhere and with finite moments of all orders. Let Qn(x), n = 0,1,2, ..., be a sequence of real valued matrix orthonormal polynomials with respect to the weight Z(x). Consider the following two Hilbert spaces: The space L2([0,1], Z(x)dx), denoted here by L2(Z), of all matrix valued measurable functions f(x), x∈[0,1], satisfying R1 0tr (f(x)Z(x)f∗(x)) dx < ∞and the space `2(M2,N0) of all real valued 2 ×2 matrix sequences (Cn)n∈N0such that P∞ n=0 tr (CnC∗ n)<∞. The map F:`2(M2,N0)−→ L2(Z) given by (An)∞ n=0 7−→ ∞ X n=0 AnQn(x) is an isometry. If the polynomials are dense in L2(Z), this map is unitary with the inverse F−1: L2(Z)−→ `2(M2,N0) given by f7−→ Aw=Z1 0 f(x)Z(x)Q∗ w(x)dx. 4 M. CASTRO AND F. A. GR ¨ UNBAUM 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 [0,1] the role of “physical space”. This is a matter of choice and one can exchange these roles. The band limiting operator, at level Nacts on `2(M2,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 T, acts on L2(Z) by multiplication by the characteristic function of the interval [0, T], T≤1. This operator will be denoted by χT. One could consider restricting time to an arbitrary subinterval (a1, b1). However, the algebraic properties exhibited here hold only with this natural restriction, and the same is true in regards to restricting the band variable. A similar situation arises in the classical case going all the way back to Shannon, [26]. Consider the problem of determining a function ffrom the following data: fhas support on the finite set {0, . . . , N}and its Fourier transform Ffis known on the compact set [0, T ]. This can be formalized as follows χTFf=g= known, χNf=f. We can combine the two equations into Ef =χTFχNf=g. To analyze this problem we need to compute the singular vectors (and values) of the operator E:`2(MR,N0)−→ L2(Z). These are given by the eigenvectors of the operators E∗E=χNF−1χTFχNand S2=EE∗=χTFχNF−1χT. The operator E∗E, acting in `2(M2,N0) is just a finite dimensional block-matrix M, and each block is given by (M)m,n = (E∗E)m,n =ZT a Qm(x)Z(x)Q∗ n(x)dx, 0≤m, n ≤N. The second operator S2=EE∗acts in L2([0, T ], Z(x)dx) by means of the integral kernel k(x, y) = N X j=0 Q∗ j(x)Qj(y). Consider now the problem of finding the eigenfunctions of E∗Eand/or EE∗. For arbitrary N and Tthere is no hope of doing this analytically, and one has to resort to numerical methods. This is a remarkably ill-conditioned problem since most of the eigenvalues are crowded together. Of all the strategies one can dream of for handling this problem, none sounds so appealing as that of finding a differential operator with simple-and spread out-spectrum which would have the same eigenfunctions as the original operators. This is exactly what Mehta as well as Slepian, Landau and Pollak did in the scalar case, when dealing with the real line and the actual Fourier transform. They discovered (the analog of) the following properties: •For each N,Tthere exists a symmetric tridiagonal matrix L, with simple spectrum, commuting with M. •For each N,Tthere exists a selfadjoint differential operator D, with simple spectrum, commuting with the integral operator S2=EE∗. TIME-AND-BAND LIMITING FOR MATRIX ORTHOGONAL POLYNOMIALS OF JACOBI TYPE 5 This paper displays one more instance where this miracle holds. Indeed there has been a systematic effort to see if the “bispectral property” first considered in [5], guarantees the commutativity of these two operators, a global and a local one. A few papers where this question has been taken up, include [8, 9, 10, 11, 13] in the scalar case as well as [1, 2, 16, 17] in the matrix valued case. It is clear that we are considering a noncommutative extension of the problem raised by C. Shannon (and solved by D. Slepian, H. Landau and H. Pollak) since these authors–as well as M. Mehta–were concerned with scalar valued functions and we are dealing with matrix valued ones. 3. The polynomials Pn(x) Our choice of the sequence Pn(x) is dictated by the simplicity of two ingredients that will be needed later on: the expression of its norm squared and its “differentiation formula”. These polynomials are certainly related to those in [24]. In particular, we will see that the entries of our choice of Pn(x) have a very “scalar hypergeometric look”. We use (a)kfor the usual Pochamer symbol. We give the entries of our Pnbelow (Pn)11(x) = n X j=0 (−x)jn j(α+β+n+ 2)j (β+ 2)j (k+j) k (Pn)12(x) = n−1 X j=0 (−x)jn j+ 1(α+β+n+ 2)j+1 (β+ 2)j+1 (j+ 1) k. (Pn)21(x) = n X j=0 (−x)jn j(α+β+n+ 3)j (β+ 2)j and finally (Pn)22(x) = n−1 X j=0 (−x)jn j(α+β+n+ 3)j (β+ 2)j+1 (β2+αβ + (n+4+j)β+ (n+ 2)α+n2+ 3n+4+j−βk −(2 + j)k (k−β−1) + (−x)n(α+β+n+ 3)n (β+ 2)n (α+β+n+ 2 −k) (k−β−1) . We have nice expressions for the (matrix valued) squared norms of the polynomials introduced above, namely 6 M. CASTRO AND F. A. GR ¨ UNBAUM hPn, Pni ≡ Z1 0 Pn(x)Z(x)P+ n(x)dx =γn     Γ(α+n+ 1)(k+n) k(α+β+ 2n+ 2)Γ(α+β+n+ 2) 0 0Γ(α+n+ 2)(k+n+ 1) (β+ 1 −k)Γ(α+β+ 2n+ 3)Γ(α+β+n+ 3)      , where γn=n!Γ(β+ 2)(α+β+n+ 2 −k) (β+ 2)n . From here we easily get a sequence of normalized polynomials, which we will denote later by Qn(x). A second advantage in the choice of our polynomials Pn(x) is that they satisfy a rather elegant looking first-order differentiation formula, namely x(1 −x)P0 n(x) = −nxPn(x) + BnPn(x) + Pn(x)Cn+DnPn−1(x) where the (xindependent) matrices Bn, Cn, Dnare given by Dn=     −n(α+n)(α+β+n+ 2 −k) (α+β+ 2n+ 1)(α+β+n+ 1 −k)−n(k+n+ 1)(β+ 1 −k) (α+β+ 2n+ 2)(k+n)(α+β+n+ 1 −k) 0−n(α+n+ 1)(k+n+ 1) (α+β+ 2n+ 2)(k+n)      , Bn=     0−(β+ 1 −k)(α+β+n+ 2) (α+β+ 2n+k)(α+β+ 2n+ 2) −(α+n+ 1)k (k+n)(α+β+n+ 2 −k)(Bn)22      , (Bn)22 =2(α+n+ 1)(α+β+n+ 2) (α+β+ 2n+ 2)(α+β+n+ 2 −k)−n(β+n+ 1) (α+β+ 2n+ 1)(α+β+ 2n+ 2)(k+n) −((α+β)2+ (3n+ 3)β+ (4n+ 3)α+ (n+ 1)(3n+ 2)) (α+β+ 2n+ 1)(α+β+ 2n+ 2) , Cn=(Cn)11 −1 0 (Cn)22, (Cn)11 =−(α+n+ 1)(α+β+n+ 2) (α+β+ 2n+ 2)(α+β+n+ 2 −k)+n(β+n+ 1) (α+β+ 2n+ 1)(α+β+ 2n+ 3)(k+n) +β+ (n+ 1)α+ (n+ 1)2 α+β+ 2n+ 1 , TIME-AND-BAND LIMITING FOR MATRIX ORTHOGONAL POLYNOMIALS OF JACOBI TYPE 7 (Cn)22 =−(α+n+ 1)(α+β+n+ 2) (α+β+ 2n+ 2)(α+β+n+ 2 −k)+n(β+n+ 1) (α+β+ 2n+ 1)(α+β+ 2n+ 2)(k+n) +n(α+n) α+β+ 2n+ 1. All of the above are the ingredients to be used to come up with normalized polynomials Qn(x) their “differentiation formula” as well as an integral kernel k(x, y) and eventually a commuting differential operator. Needless to say these last two ingredients depend on the two parameters going along with “time” and “band” limiting respectively, i.e. Tand N. 4. The commuting differential operator As mentioned earlier, at the end of the introduction, we postulate for the commuting differential operator ˜ Dthe following one ˜ D= (x−T)D−d dxx(1 −x) + PN(x) where Dwas given earlier and PN(x) = x(N2+ (α+β+ 3)(N+ 1)) α+β+N+ 2 x(k−β−1) x(N2+ (α+β+ 4)N+ 2α+ 2β−k+ 6) + β. As discussed extensively in [2, 17], to establish this commutativity we need to show that (k∗(x, y)) ˜ Dx= (k(x, y)˜ Dy)∗ where ˜ Dxand ˜ Dydenote the action of ˜ Dabove on the xand yvariables, respectively. Given the expression for Dwe get that ˜ Dxf= 2 d dxfx(1 −x)+(x−T)d2 dx2fx(1 −x) +f(C−xU)+(x−T)d dxf(C−xU)−(x−T)fV −df dxx(1 −x) + fPN(x) = (x−T)fD +d dxfx(1 −x) + f(PN(x) + C−xU). We are thus claiming that the expression 8 M. CASTRO AND F. A. GR ¨ UNBAUM k∗(x, y)˜ Dx=  N X j=0 Q∗ j(y)Qj(x) ˜ Dx = (x−T) N X j=0 Q∗ j(y)∧jQj(x) + x(1 −x) N X 0 Q∗ j(y)Q0 j(x) +  N X j=0 Q∗ j(y)Qj(x) (PN(x) + C−xU) agrees with (k(x, y)˜ Dy)∗= (y−T) N X j=0 Q∗ j(y)∧jQj(x) +y(1 −y) N X j=0 Q∗ j 0(y)Qj(x) + (P∗ N(y) + C∗−yU∗) N X 0 Q∗ j(y)Qj(x). The key ingredients to go further are the differentiation formula mentioned earlier and the Christoffel–Darboux formula given below, see [4]. This strategy has been used since the early work in [8]. Both of them will involve the polynomials Qj(x). We note that if hPj, Pji ≡    1 σ2 1(j)0 01 σ2 2(j)   , we have Qj(x) = σ1(j) 0 0σ2(j)Pj(x)≡σ(j)Pj(x), where σ1(j) and σ2(j) have nice explicit expressions given earlier. The differentiation formula becomes x(1 −x)Q0 j(x) = −jxQj(x) + σjBjσ−1 jQj(x) + Qj(x)Cj+σjDjσ−1 j−1Qj−1(x), and we will use later on the expressions ˜ Bj=σjBjσ−1 j,˜ Dj=σjDjσ−1 j−1. The Christoffel–Darboux formula requires knowing a part of the three-term recursion satisfied by the polynomials Pj(x), and eventually for the Qj(x) themselves. TIME-AND-BAND LIMITING FOR MATRIX ORTHOGONAL POLYNOMIALS OF JACOBI TYPE 9 One easily gets xPj(x) = FjPj−1(x) + GjPj(x) + HjPj+1(x) where the expression for Fjis particularly simple, namely Fj=      −j(α+j)(k−α−β−2−j) (α+β+j+ 1)(α+β+ 2j+ 2)(k−α−β−1−j) j(1 + β−k) (α+β+ 2j+ 2)(k+j)(k−α−β−1−j) 0−j(α+1+j)(k+1+j) (α+β+ 2j+ 2)(α+β+ 2j+ 3)(k+j)       . It follows that the polynomials Qj(x) satisfy the recursion xQj(x) = ˜ FjQj−1(x) + ˜ GjQj(x) + ˜ HjQj+1(x) with ˜ Fj=σjFjσ−1 j−1,˜ Gj=σjGjσ−1 j,˜ Hj=σjHjσ−1 j+1. The Christoffel–Darboux identity states that N X j=0 Q∗ j(y)Qj(x) = Q∗ N(y)˜ F∗ N+1QN+1(x)−Q∗ N+1(y)˜ FN+1QN(x) x−y. With these ingredients at hand one can go for a proof along the lines of [2, 8, 17] or one can try as an alternative a proof by induction of the value of N, as discussed below. For N= 0 the identity in question can be checked explicitly. Assuming that this has been proved up to N=j, one can look at the increment in the left-hand side and the right-hand side of the identity we are trying to prove, i.e. lhs(j+ 1) −lhs(j) and rhs(j+ 1) −rhs(j). Using both the differentiation formula and the Christoffel–Darboux identity the difference of both increments, one on the left hand side and one on the right hand side, is given by (x−y)Q∗ j+1(y)∧j+1 Qj+1(x)+ {Q∗ j+1(y)(−(j+ 1)xQj+1(x) + ˜ Bj+1Qj+1(x) + Qj+1(x)˜ Cj+1 +˜ Dj+1Qj(x)) −(−(j+ 1)yQ∗ j+1(y) + Q∗ j+1(y)˜ B∗ j+1 +˜ C∗ j+1Q∗ j+1(y) + Q∗ j(y)˜ D∗ j+1)Qj+1(x)} +(Q∗ j(y)˜ F∗ j+1Qj+1(x)−Q∗ j+1(y)˜ Fj+1Qj(x) x−y(Pj+1(x)− Pj(x)) +Q∗ j+1(y)Qj+1(x)(Pj+1(x) + C−xU) −((P∗ j+1(y)− P∗ j(y))Q∗ j(y)˜ F∗ j+1Qj+1(x)−Q∗ j+1(y)˜ Fj+1Qj(x) x−y + (P∗ j+1(y) + C∗−yU∗)Q∗ j+1(y)Qj+1(x).