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On the inversion of infinite moment matrices
Linear Algebra and its Applications 475 (2015) 292–305
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Linear Algebra and its Applications www.elsevier.com/locate/laa
On the inversion of infinite moment matrices C. Escribano 1 , R. Gonzalo ∗,1 , E. Torrano 1 Departamento de Matemática Aplicada, Facultad de Informática de Madrid, Universidad Politécnica, Campus de Montegancedo, Boadilla del Monte, 28660, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 16 September 2014 Accepted 24 February 2015 Available online 10 March 2015 Submitted by P. Semrl MSC: 44A60 15A29 Keywords: Hermitian moment problem Orthogonal polynomials Smallest eigenvalue Measures Inverses of infinite matrices
a b s t r a c t Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: cescribano@fi.upm.es (C. Escribano), rngonzalo@fi.upm.es (R. Gonzalo), emilio@fi.upm.es (E. Torrano). 1 Tel.: +34 913367419; fax: +34 913367426. http://dx.doi.org/10.1016/j.laa.2015.02.031 0024-3795/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction Let M = (ci,j )∞ i,j=0 be an infinite Hermitian matrix, i.e., ci,j = cj,i for all i, j nonnegative integers. Following [12] we say that M is positive definite (in short, an HPD matrix) if |Mn | > 0 for all n ≥ 0, where Mn is the truncated matrix of size (n +1) ×(n +1) of M. To each HPD matrix M can be associated an inner product on the linear space of polynomials P[z] as follows: if p(z) = n vn z n and q(z) = n wn z n then ⎞ w0 ⎟ ⎜ ⎜ w1 ⎟ ⎜ ... M⎜w ⎟ ⎟. ⎝ 2⎠ .. . ⎛
p, q =
v0
v1
v2
(1)
M is the Gram matrix of the inner product (1) in the vector space of polynomials P[z], i.e., < z i , z j >= ci,j . Let {Pn (z)}∞ n=0 be the orthonormal polynomials with respect to such inner product and write Pn (z) =
n
bk,n z k .
k=0
The orthonormal polynomials {Pn (z)}∞ n=0 are uniquely determined by orthonormality if bn,n > 0 and can be given as (see e.g. [19])
c0,0
c
1,0
. 1
. Pn (z) = . |Mn−1 ||Mn |
cn−1,0
1
c0,1 c1,1 .. . cn−1,1 z
, n ≥ 1,
. . . cn−1,n
... zn ... ...
c0,n c1,n .. .
P0 = 1.
(2)
Define the infinite upper triangular matrix B = (bk,n )∞ k,n=0 with bk,n = 0 if k > n. This matrix can be considered as the infinite transition matrix from the algebraic basis n ∞ B = {Pn (z)}∞ n=0 to B = {z }n=0 . Indeed, each finite section Bn of B is the corresponding transition matrix from the algebraic basis Bn = {P0 (z), . . . , Pn (z)} in the linear vector space Pn [z] of polynomials of degree at most n, to the standard basis Bn = {1, z, . . . , z n }. Since Mn , In are both matricial representations of the same inner product with respect to Bn , Bn respectively, then Btn Mn Bn = In , t and consequently M−1 n = Bn Bn . This result in the case of Hankel matrices going back to A.C. Aitken, cf. [10] has been recovered several times see [5,8] (also [12] in the context of moment Hermitian matrices). In particular, if An = Bn Btn , it follows that
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Mn An = An Mn = In .
(3)
A natural question related to the above equality is to know if (3) remains true for infinite matrices in the particular case that the matrix multiplication A = BBt is well defined, that is, if it is true that A is the classical inverse of M in the sense that MA = AM = I.
(4)
The purpose of the present paper is to study this problem in the general framework of HPD matrices or in the particular case of Toeplitz matrices, and some questions related with it. In this direction in [8] the authors studied this problem in the particular case of positive definite Hankel matrices H which are moment matrices associated with measures supported in the real line. They stated that it is likely that the answer is positive in the indeterminate case or equivalently, as a consequence of [5], when limn→∞ λn > 0, where λn is the smallest eigenvalue of the finite section Hn of H. Our approach is also from the point of view of the asymptotic behaviour of the smallest eigenvalues λn of the finite sections of the HPD matrices. We prove that the answer is negative in the case of Toeplitz matrices, see Example 1. Our results do not contribute to the question about Hankel matrices. Recall that an HPD matrix M is the moment matrix associated with a Borel measure μ with support on the complex plane C if for all i, j ≥ 0 z i z j dμ.
ci,j =
In this case we denote M = M(μ). Note that the inner product (1) induced by M(μ) in P[z] is the inner product in L2 (μ), indeed, if p(z), q(z) ∈ P[z] p, q =
p(z)q(z)dμ.
For more information concerning the characterization of HPD matrices which are moment matrices with respect to a certain measure μ with support on C see among others [2,7,18]. Since moment matrices associated with measures with infinite support on C are HPD matrices, many of the examples that appear in this paper are indeed moment matrices. Note that in this particular case the corresponding orthonormal polynomials with respect to μ and having positive leading coefficients bn,n (μ) are given as Pn (z) =
n
bk,n (μ)z k .
k=0
In the context of the theory of orthogonal polynomials our aim is to study if we can compute the classical inverse of a moment matrix in terms of the coefficients of the orthonormal polynomials. For finite matrices the approach of determining the inverse of
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a finite Hankel or Toeplitz matrix in connection with the theory of orthogonal polynomials appears in [21] and [22] where an algorithm for the inversion of such matrices is obtained. Later on, this point of view is also treated in [16] for certain finite moment Hankel matrices; in the finite dimensional case this kind of algorithms enables to compute inverse matrices faster. We point out that our results are for infinite dimensional Hermitian matrices. The paper is organized as follows: in Section 2 we provide a counterexample showing that the matrix A is not, in general, a classical inverse of an HPD matrix M when limn→∞ λn > 0. However, our example is not a Hankel matrix, and in this case the problem remains open. We show that in the particular case of HPD matrices with square summable rows the condition limn→∞ λn > 0 is sufficient to assure that A is a classical inverse of M. This condition is not a necessary condition as we show. Section 3 is devoted to study the main problem in the context of HPD Toeplitz matrices. In this direction, using the persymmetric property of the inverse of a finite Toeplitz matrix we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the classical inverse of an HPD Toeplitz matrix is an asymptotic Toeplitz matrix. Moreover, since every HPD Toeplitz matrix is a moment matrix associated with a measure on the unit circle T (see e.g. [1]), we give the description of the classical inverse of an HPD Toeplitz matrix in terms of the limits of the coefficients of the orthonormal polynomials. We apply our techniques to the inversion of Toeplitz operators not necessarily bounded. As a consequence we obtain that the asymptotic limit of the inverse of a Toeplitz matrix Tϕ associated with a continuous symbol verifying inf z∈T ϕ(z) > 0 is the Toeplitz matrix T ϕ1 . Finally, in Section 3 we also obtain some applications of the preceding results for orthonormal polynomials. In particular, we show that whenever limn→∞ λn (ν) > 0 we may assure the existence of all the limits of the coefficients of the orthonormal polynomials limn→∞ bn−k,n (ν) for every k ≥ 0. Note that in the case of the main coefficients the existence of limn→∞ bn,n (ν) was already known by Szegö theory; the case of the other coefficients, up to our knowledge, was not known. First, we introduce some notation. For A = (ai,j )ni,j=0 being a finite matrix we identify the linear operator on Cn+1 induced by A with its matrix with respect to the standard basis of Cn+1 . However in the infinite case we distinguish between infinite matrices and operators using different ∞ notation. If A is an infinite matrix and the matrix product ∞ t Ax = exists for every x ∈ 2 and belongs to 2 , then the graph is j=0 ai,j xj i=0 closed and by the closed graph theorem A defines a bounded operator that we will denote by A. Note that there are infinite matrices not defining bounded operators, consider for t example the diagonal matrix A = (iδi,j )∞ i,j=0 , since the product Ax exists for all x ∈ 2 but the result does not always belong to 2 . In [11] a criterion has been proved to characterize when an infinite matrix defines a bounded operator from 2 to 2 . On the other hand, it is well known that if A : H → H is a bounded operator on a countable Hilbert space H with an orthonormal basis B = {vi }∞ i=0 then the associated matrix is A = (Avj , vi )∞ . i,j=0
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2. Infinite transition matrices. Inversion of certain HPD matrices Let M be an HPD matrix and let B be the transition matrix associated with M. t Let A = (ai,j )∞ such formal matrix product is i,j=0 be the matrix A = BB whenever ∞ well defined, i.e. if for all i, j ∈ N0 there exists ai,j = k=max{i,j} bi,k bj,k . In the case of M = M(μ) being a moment matrix associated with a measure μ we denote by A = A(μ). Remark 1. Note that the existence of A = BBt is equivalent to all the rows of the matrix B being square summable. One can ask if the matrix B comes from a bounded operator in a Hilbert space with a countable orthonormal basis like 2 or the completion of the polynomials with the inner product from M. In a paper of Berg–Durán [6] the authors consider an operator T on the polynomials given by T (z n ) = Pn . Since T (Pn ) =
n
bk,n Pk
k=0
the operator T has the matrix B with respect to the orthonormal basis {Pn (z)}∞ n=0 . Then Theorem 1.1 in [6] can be reformulated as: Lemma 1. Let M be an infinite HPD matrix and let B be the transition matrix associated with M. Then, the following are equivalent: (1) limn→∞ λn = λ > 0. (2) The matrix B defines a bounded operators B on 2 and B = λ−1/2 . As a consequence of (1) we give a sufficient condition in terms of the asymptotic behaviour of λn to assure the existence of the matrix A defining a bounded operator A on 2 . Lemma 2. Let M be an HPD matrix such that limn→∞ λn = λ > 0, then the matrix ∗ A = BBt = (ai,j )∞ i,j=0 exists and defines the bounded operator A = BB on 2 . Moreover, (1) ai,j = limn→∞ M−1 n [i, j] for every i, j ∈ N0 . (2) A = B2 = λ−1 . Proof. By Lemma 1 the matrices B, Bt define bounded operators on 2 and consequently the rows and columns of such matrices belong to 2 . Then for every i, j ∈ N0 the series ∞ k=max{i,j} bi,k bj,k is absolutely convergent and ai,j =
∞
bi,k bj,k = lim
n
n→∞
k=max{i,j}
bi,k bj,k = lim M−1 n [i, j]. n→∞
k=max{i,j}
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Therefore the matrix A exists and is the representation with respect to the standard ∗ ∗ basis of 2 of the operator BB . Moreover,A = BB 2 = B2 = λ−1 . 2 Remark 2. In the case limn→∞ λn = 0 we cannot assure even the existence of the matrix A, as the following example shows: consider the Lebesgue measure μ on the circle with center (1, 0) and radius 1, and the associated moment matrix ⎛
1 ⎜1 ⎜ ⎜ 1 M(μ) = ⎜ ⎜ ⎜1 ⎝ .. .
1 1 1 2 3 4 3 6 10 4 10 20 .. .. .. . . .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .
The sequence of orthonormal polynomials associated with μ is Pn (z) = (z − 1)n for all ∞ n ≥ 0 and since k=0 |b0,k (μ)|2 = ∞ it follows that A(μ) does not exist. Note that n−k n the transition matrix is B(μ) = (bk,n (μ))∞ k,n=0 with bk,n (μ) = (−1) k if k ≤ n and bk,n = 0 if k > n which obviously does not define a bounded operator on 2 . We show that the answer of the main problem is negative for HPD matrices: Example 1. There exists a Toeplitz HPD matrix T such that limn→∞ λn > 0 and the matrix A is not a classical inverse of T, i.e., TA = I,
AT = I.
Indeed, consider the moment matrix T(ν) associated with the measure ν = 12 m + μ0 where m is the Lebesgue measure on T and μ0 is the atomic measure supported on the point 1 and with μ0 ({1}) = 12 , that is ⎛
1
⎜ 12 ⎜1 ⎜ T(ν) = ⎜ 2 ⎜1 ⎝2 .. .
1 2
1 1 2 1 2
.. .
1 2 1 2
1 1 2
.. .
1 2 1 2 1 2
1 .. .
⎞ ... ...⎟ ⎟ ...⎟ ⎟. ...⎟ ⎠ .. .
Since T(ν) ≥ 12 I in the sense that for every v ∈ c00 we have vT(ν)v ∗ ≥ 12 vv ∗ , it follows that λn (ν) ≥ 12 for all n ∈ N0 and limn→∞ λn (ν) ≥ 12 . It can be checked that 2(n+1) 2 −1 M−1 n [i, j] = − n+2 if i = j and Mn [i, i] = n+2 for each n ∈ N0 and 0 ≤ i, j ≤ n. By using Lemma 2 it follows that ai,j = limn→∞ M−1 n [i, j] = 2δi,j and A(ν) = 2I. Therefore A(μ)T(ν) = I
and
T(ν)A(ν) = I.
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Using the efficient numerical algorithms of Cholesky decomposition and inversion of lower triangular matrices we can determine the explicit representation of the transition matrix √ √ √ 2n and bk,n = − 2 B(ν) = (bk,n )∞ with b = if k < n. n,n k,n=0 n+1 n(n+1) In the following proposition we prove that the condition limn→∞ λn > 0 is a sufficient condition for A being the inverse of an HPD matrix M in the particular case of M having square summable rows. Remark 3. Let M = (ci,j )∞ i,j=0 be an HPD matrix with square summable rows, i.e. ∞ 2 j=0 |ci,j | < ∞ for all i ≥ 0; then the columns are also square summable. Consider c00 the subspace of 2 that consists of all complex sequences with only finitely many non-zero entries. It is clear that Metk ∈ 2 being {ek } the standard basis of 2 , consequently Mxt ∈ 2 for every x ∈ c00 . Therefore the linear mapping M : c00 → 2 given by the formal matrix product Mxt is well defined, not necessarily defining a bounded operator. Proposition 1. Let M = (ci,j )∞ i,j=0 be an HPD matrix with square summable rows. Assume that limn→∞ λn > 0 then AM = MA = I. Proof. First of all since M is Hermitian it has square summable rows and columns. By Lemma 2 since limn→∞ λn > 0 it follows that the matrix A is the matrix representation of a bounded operator on 2 and, consequently, the rows and columns of A belong to 2 . Then both matrices AM and MA are well defined. We first show that for each j, k ∈ N0 AM[j, k] = δj,k . We introduce the following notation: for every n ≥ 0 denote by Πn the projection Πn : 2 → 2 defined by Πn (x0 , x1 , . . . , xn , . . .) = (x0 , x1 , . . . , xn , 0, 0, . . .) for every x ∈ 2 . Let B be a bounded on 2 , we denote by B˜n = Πn BΠn , which matrix representation operator Bn 0 is given by . It is well known (see e.g. [14]) that {B˜n }∞ n=0 is strongly convergent 0 0 ˜ }∞ and {B˜∗ }∞ are strongly convergent to B, B∗ respectively, to B on 2 . Since {B n n=0 n n=0 ∗ ˜∗ ∞ ˜ then {Bn Bn }n=0 is strongly convergent to BB . Fix k ∈ N0 , since (ci,k )∞ i=0 ∈ 2 it follows that for every n ≥ k we have that ⎛ AMetk
= lim
n→∞
Bn B∗n 0
c0,k c1,k .. . cn,k
⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ Bn B∗n ⎟ ⎜ ⎟ = lim n→∞ ⎜ ⎟ 0 0 ⎜ ⎟ ⎜ cn+1,k ⎟ ⎝ ⎠ .. .
⎛
⎞ c0,k ⎜c ⎟ ⎜ 1,k ⎟ ⎜ . ⎟ . ⎟ 0 ⎜ ⎜ . ⎟ ⎜ ⎟ = ek . 0 ⎜ cn,k ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎝ ⎠ .. .
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Therefore ej AMetk = δj,k . On the other hand, since M and A are Hermitian matrices MA = I.
2
Remark 4. Unfortunately Proposition 1 does not provide information in the context of Hankel matrices since if limn→∞ λn > 0 the Hankel matrix H = (si+j )∞ i,j=0 does not ∞ have square summable rows. Indeed, if n=0 s2n < ∞ it follows limn→∞ sn = 0. On the other hand, for every n ∈ N0 , s2n = e2n He∗2n ≥ λn and therefore limn→∞ λn = 0. Remark 5. The condition limn→∞ λn > 0 is not necessary for A being the classical inverse of an HPD matrix with square summable rows, even in the case of moment matrices as ∞ the following example shows: let 0 < a < 1 and M = amax{i,j} i,j=0 , that is ⎛
1 ⎜ a ⎜ ⎜ 2 a M=⎜ ⎜ 3 ⎜a ⎝ .. .
a a a2 a3 .. .
a2 a2 a2 a3 .. .
a3 a3 a3 a3 .. .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. . n(n+1)
It can be easily checked that M is positive definite since |Mn | = a 2 (1 − a)n > 0 ∞ for each n ∈ N0 . Consider the diagonal matrix D = ai/2 δi,j i,j=0 and the matrix |i−j| ∞ 1 Pr (θ) T= a 2 which is the Toeplitz matrix (see [17]) for the Poisson kernel 2π i,j=0 √ with r = a and Pr (θ) =
∞
r|n| einθ .
n=−∞
Then it holds that M = Dt TD. Taking in account this equality it is obvious that T is an HPD Toeplitz matrix and consequently T = T(ν) for a certain measure ν with support on T. Using [13] and [15] it follows that M is the moment matrix of the image √ measure μ = ν ◦ f −1 under the transformation f (z) = az. In this particular case it is easy to obtain the orthonormal polynomials by (2)
1
a 1
. Pn (z) = |Mn−1 ||Mn |
..
1 Consequently,
a . . . an
a . . . an z n−1 (z − a) , n = 1, 2, . . . .. ..
= n a (1 − a) . .
z . . . zn
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⎛
1
⎜ ⎜0 ⎜ ⎜ 0 B(μ) = ⎜ ⎜ ⎜0 ⎜ ⎝ .. .
−a a(1−a) 1 a(1−a)
0
0
0
−a a2 (1−a) 1 a2 (1−a)
0 .. .
0 .. .
...
0 −a a3 (1−a) 1 a3 (1−a)
.. .
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎟ ⎠ .. .
In particular, B(μ) does not define a bounded operator on 2 and by Lemma 1 we have that limn→∞ λn (μ) = 0. On the other hand it can be checked that A(μ)M(μ) = M(μ)A(μ) = I, where ⎛
1 ⎜ −1 ⎜ 1 ⎜ ⎜ 0 A(μ) = 1−a⎜ ⎜ 0 ⎝ .. .
−1 a+1 a −1 a
0 .. .
0 −1 a a+1 a2 −1 a2
.. .
0 0 −1 a2 a+1 a3
.. .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .
3. Inversion of Toeplitz HPD matrices It is well known that the inverse of a Toeplitz matrix is not, in general, a Toeplitz matrix, even in the finite dimensional case. Nevertheless, such inverse is persymmetric as explained below. For an (n + 1) × (n + 1) matrix A = (ai,j )ni,j=0 introduce the matrix Aπ by Aπ [i, j] = A[n − j, n − i] = an−j,n−i ,
i, j = 0, . . . , n
Then Aπ = JAt J, where J[k, l] = 1 if and only if k + l = n and otherwise J[k, l] = 0 (i.e. zeros except in the south-west to north-east diagonal having ones). Clearly (AB)π = Bπ Aπ , so if A is regular then so is Aπ , and in the affirmative case (Aπ )−1 = (A−1 )π . The matrix A is called persymmetric iff A = Aπ , i.e., if and only if it is symmetric with respect to the south-west to north-east diagonal. A Toeplitz matrix T is clearly persymmetric. Therefore if A is regular and persymmetric then A−1 is persymmetric. In particular, if T is a regular Toeplitz matrix, then T−1 is persymmetric. We show that this property has a great impact on the form of a classical inverse of certain infinite Toeplitz matrices. In order to do it we introduce the notion of asymptotic Toeplitz matrix very closely related to the notion of asymptotic Toeplitz operator that appears in [4]; indeed, in the particular case of matrices defining bounded operators in
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Hilbert spaces, these matrices are the representations of such operators with respect to orthonormal basis. Definition 1. An infinite matrix A = (ai,j )∞ i,j=0 is an asymptotic Toeplitz matrix if for every k ∈ Z there exists limn→∞ an,n+k = αk . In such a case we denote by Lim(A) = (αi−j )∞ i,j=0 which is clearly a Toeplitz matrix. The main result in this section is: Theorem 1. Let T be an infinite HPD Toeplitz matrix and let B = (bk,n )∞ k,n=0 be the transition matrix associated with T. Assume that the matrix A = BBt = (ai,j )∞ i,j=0 exists. Then: (1) B is an asymptotic Toeplitz matrix, i.e. for every k limn→∞ bn−k,n = βk . (2) For i, k ∈ N0 with i ≤ k, ai,k =
i
∈
N0 there exists
βj βk−j .
j=0
(3) Moreover, if
∞ k=0
βk2 < ∞ then the matrix A is an asymptotic Toeplitz and lim ak,k+j =
k→∞
∞
βi βi+j ,
i=0
where the series above is absolutely convergent. Proof. We first show that there exists limn→∞ bn,n . Indeed, since A exists and T−1 n = Bn Btn a0,0 =
∞ k=0
|b0,k |2 = lim
n→∞
n k=0
|b0,k |2 = lim T−1 n [0, 0]. n→∞
−1 Using that the matrix Tn−1 is persymmetric it follows that T−1 n [0, 0] = Tn [n, n] and therefore 2 a0,0 = lim T−1 n [n, n] = lim bn,n . n→∞
n→∞
√ Moreover β0 = limn→∞ bn,n = a0,0 > 0 since bn,n > 0 for each n ∈ N0 . Now we prove the existence of limn→∞ bn−k,n for every k ∈ N. Indeed, using again that T−1 n is persymmetric −1 a0,k = lim T−1 n [0, k] = lim Tn [n − k, n] = lim bn−k,n bn,n . n→∞
n→∞
n→∞
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Since limn→∞ bn,n = β0 > 0 we have that there exists limn→∞ bn−k,n and moreover, lim bn−k,n =
n→∞
a0,k . β0
In a general way we can determinate all of the entries of the matrix A. Indeed, let i, k ∈ N0 be fixed with i ≤ k then −1 t T−1 n [i, k] = Tn [n − k, n − i] = (Bn Bn )[i, k] =
i
bn−i,n−i+j bn−k,n−i+j .
j=0
By passing to the limit and using Lemma 2 ⎛
ai,k = lim T−1 n [i, k] = n→∞
i
⎞ βk−i ⎟ ⎜ ⎜ ... ⎟ ⎜ ⎟. · · · βi ⎜ ⎟ ⎝ βk−1 ⎠ βk
β j βk+j−i =
β0
β1
j=0
Thus, we have the description of A. Note that the entries of the main diagonal of A are k given by ak,k = i=0 βi2 . In order to prove (3) let j ∈ N0 be fixed and k ∈ N, then ⎛
ak,k+j =
The series exists
∞ i=0
β0
β1
⎞ βk+j−k ⎟ k .. ⎜ ⎜ ⎟ . ⎜ ⎟= βi βi+j . · · · βk ⎜ ⎟ ⎝ βk−1 ⎠ i=0 βk+j
βi βi+j is absolutely convergent since
lim ak,k+j =
k→∞
∞
β i βi+j .
∞ k=0
βk2 < ∞, and therefore there
2
i=0
Remark 6. Note that in Example 1 the corresponding matrix B(ν) associated with ν is √ asymptotic Toeplitz and limn→∞ bn−k,n (ν) = 0 if k > 0 and limn→∞ bn,n (ν) = 2. As a consequence of the results of the preceding section we obtain the explicit representation of the inverse of an HPD Toeplitz matrix with limn→∞ λn > 0 in terms of the coefficients of the orthonormal polynomials. The existence of the classical inverse in this case was known (see e.g. [9]) in the context of the nonhomogeneous solutions of systems of infinite equations.
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Corollary 1. Let T = (ci−j )∞ i,j=0 be an HPD Toeplitz matrix with limn→∞ λn > 0 and ∞ 2 j=0 |cj | < ∞. Then the matrix A is the classical inverse of T and verifies that for i, k ∈ N0 with i ≤ k, i
ai,k =
βj βk−j ,
j=0
where βk = limn→∞ bn−k,n for k ≥ 0. The main consequence of Theorem 1 in the context of the orthogonal polynomials is that, in the case of limn→∞ λn > 0, we may assure the existence of the limits of the coefficients of the orthonormal polynomials associated with a measure with support on T. Corollary 2. Let ν be a measure with in infinite support on T and let Pn (z) = n k the orthonormal polynomial with respect to ν. Assume that k=0 bk,n (ν)z limn→∞ λn > 0, then for every k ∈ N0 there exists (is finite) the following limit: lim bn−k,n (ν).
n→∞
Moreover, 0 < limn→∞ bn,n (ν) < ∞. Remark 7. We may rewrite the result above to obtain a consequence in the asymptotic (1) (2) ∞ behaviour of the connection coefficients. Recall that if {Qn }∞ n=0 and {Qn }n=0 are two system of monic polynomials corresponding to the measures μ1 and μ2 and Q(2) n (z) =
n
c(k, n)Q(1) n (z),
k=0 (1)
(2)
we call {c(k, n)}k,n the connection coefficients from {Qn } to {Qn } (see e.g. [20]). In particular, if ν is a measure on T and m is the Lebesgue measure, since {z n }∞ n=0 is the sequence of monic polynomials with respect to m we may rewrite the system of monic polynomials with respect to ν as: Qn (z) =
n
c(k, n, ν)z k
k=0
bk,n (ν) and c(n, n, ν) = 1. As a consequence of the results in the bn,n (ν) preceding section we have: where c(k, n, ν) =
Corollary 3. Let ν be a measure with infinite support on T and let {Qn (z)}∞ n=0 be the corresponding sequence of monic orthogonal polynomials given by:
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Qn (z) =
n
c(k, n, ν)z k .
k=0
If limn→∞ λn (ν) > 0, then for every k ∈ N0 there exist (are finite) lim c(n, n + k, ν)
n→∞
Proof. Note that if limn→∞ λn (ν) > 0 then the matrix A(ν) exists and therefore by Theorem 1 the matrix B(ν) is asymptotic Toeplitz, and moreover lim bn,n (ν) = κ(ν) > 0
n→∞
Then, we may conclude that for every k ∈ N0 there exists (is finite) the following limit: bn,n+k (ν) . n→∞ bn,n (ν)
lim c(n, n + k, ν) = lim
n→∞
2
We finish this section with some consequences of the inversion result of Toeplitz matrices defining, in general, not necessarily bounded Toeplitz operators. Corollary 4. Let Tϕ = (ci−j )∞ i,j=0 be a Toeplitz HPD matrix associated with a symbol ϕ ∈ L2 (T) and ess inf ϕ(z) > 0. Then the matrix Aϕ = BBt exists and is the classical inverse matrix of Tϕ , i.e. Aϕ Tϕ = Tϕ Aϕ = I. In the particular case of bounded Toeplitz matrices we obtain: Proposition 2. Let Tϕ be an HPD Toeplitz matrix with continuous symbol ϕ such that inf ϕ(z) > 0 then Aϕ is asymptotic Toeplitz and Lim(Aϕ ) = T ϕ1 . 1 ϕ ϕ, ϕ1
Proof. Since over, since
is continuous on T then T ϕ1 defines a bounded Toeplitz operator. More∈ L∞ (T) by [4] it follows that
I − Tϕ T ϕ1 = Hzϕ Hz ϕ1 = K Now, by [3] since the symbols zϕ and z ϕ1 are continuous then the Hankel matrices Hzϕ , Hz ϕ1 define compact operators and consequently K defines a compact operator. Therefore K is an asymptotic matrix with Lim(K) = 0. Since A defines a bounded operator which is the inverse operator of Tϕ then Tϕ (A − T ϕ1 ) = Tϕ A − Tϕ T ϕ1 = K, and A − T ϕ1 = ATϕ (A − T ϕ1 ) = AK.
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Since A defines a bounded operator and K defines a compact operator then the matrix AK is the matrix representation of a compact operator and therefore such matrix is weakly asymptotically Toeplitz with limit 0 and we have the conclusion. 2 Acknowledgements This work has been partially supported by the project TEC2012-35673, funded by Ministerio de Economía y Competitividad, Spain. References [1] N.I. Akhiezer, The Classical Moment Problem, Oliver and Boyd Ltd., Edinburgh and London, 1965. [2] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975) 317–325. [3] R.A. Martinez-Avendaño, P. Rosenthal, An Introduction to Operators on the Hardy–Hilbert Space, Graduate Texts in Mathematics, vol. 237, 2007. [4] J. Barriá, P.R. Halmos, Asymptotic Toeplitz operators, Trans. Amer. Math. Soc. 273 (1982) 621–630. [5] C. Berg, Y. Chen, M.E.H. Ismail, Small eigenvalues of large Hankel matrices: the indeterminate case, Math. Scand. 91 (2002) 67–81. [6] C. Berg, A.J. Durán, Orthogonal polynomials and analytic functions associated to positive definite matrices, J. Math. Anal. Appl. 315 (2006) 54–67. [7] C. Berg, P. Maserick, Exponentially bounded positive definite functions, Illinois J. Math. 28 (1984) 162–179. [8] C. Berg, R. Szwarc, The smallest eigenvalue of Hankel matrices, Constr. Approx. 34 (2011) 107–133. [9] M. Bernkopf, A history of infinite matrices, Arch. Hist. Exact Sci. 4 (1968) 308–358. [10] A.R. Collar, On the reciprocation of certain matrices, Proc. R. Soc. Edinb. (1939) 301–312. [11] L. Crone, A characterization of matrix operators on 2 , Math. Z. 123 (1991) 315–317. [12] C. Escribano, R. Gonzalo, E. Torrano, Small eigenvalues of large Hermitian moment matrices, J. Math. Anal. Appl. (2010). [13] C. Escribano, M.A. Sastre, E. Torrano, Moment matrix of self-similar measures, Electron. Trans. Numer. Anal. 24 (2006) 79–87. [14] P.R. Halmos, A Hilbert Space Problem Book, second edition, Springer-Verlag, New York, 1980. [15] P.E.T. Jorgensen, K.A. Kornelson, K.L. Shuman, Iterated Function Systems, Moments, and Transformations of Infinite Matrices, Memoirs of the American Mathematical Society, vol. 213 (1003), 2011. [16] S.B. Provost, H. Ha, On the inversion of certain moment matrices, Linear Algebra Appl. 430 (2009) 2650–2658. [17] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987. [18] F.H. Szafraniec, Boundedness of the shift operator related to definite positive forms: an application to moment problem, Ark. Math. 19 (1981) 251–259. [19] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 32, 1939. [20] R. Szwarc, Connection coefficients of orthogonal polynomials, Canad. Math. Bull. 35 (4) (1992) 548–856. [21] W.F. Trench, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Ind. Appl. Math. 12 (1964) 515–522. [22] W.F. Trench, An algorithm for the inversion of finite Hankel matrices, J. Soc. Ind. Appl. Math. 13 (4) (1965) 1102–1107.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On the inversion of infinite moment matrices C. Escribano 1 , R. Gonzalo ∗,1 , E. Torrano 1 Departamento de Matemática Aplicada, Facultad de Informática de Madrid, Universidad Politécnica, Campus de Montegancedo, Boadilla del Monte, 28660, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 16 September 2014 Accepted 24 February 2015 Available online 10 March 2015 Submitted by P. Semrl MSC: 44A60 15A29 Keywords: Hermitian moment problem Orthogonal polynomials Smallest eigenvalue Measures Inverses of infinite matrices
a b s t r a c t Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: cescribano@fi.upm.es (C. Escribano), rngonzalo@fi.upm.es (R. Gonzalo), emilio@fi.upm.es (E. Torrano). 1 Tel.: +34 913367419; fax: +34 913367426. http://dx.doi.org/10.1016/j.laa.2015.02.031 0024-3795/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction Let M = (ci,j )∞ i,j=0 be an infinite Hermitian matrix, i.e., ci,j = cj,i for all i, j nonnegative integers. Following [12] we say that M is positive definite (in short, an HPD matrix) if |Mn | > 0 for all n ≥ 0, where Mn is the truncated matrix of size (n +1) ×(n +1) of M. To each HPD matrix M can be associated an inner product on the linear space of polynomials P[z] as follows: if p(z) = n vn z n and q(z) = n wn z n then ⎞ w0 ⎟ ⎜ ⎜ w1 ⎟ ⎜ ... M⎜w ⎟ ⎟. ⎝ 2⎠ .. . ⎛
p, q =
v0
v1
v2
(1)
M is the Gram matrix of the inner product (1) in the vector space of polynomials P[z], i.e., < z i , z j >= ci,j . Let {Pn (z)}∞ n=0 be the orthonormal polynomials with respect to such inner product and write Pn (z) =
n
bk,n z k .
k=0
The orthonormal polynomials {Pn (z)}∞ n=0 are uniquely determined by orthonormality if bn,n > 0 and can be given as (see e.g. [19])
c0,0
c
1,0
. 1
. Pn (z) = . |Mn−1 ||Mn |
cn−1,0
1
c0,1 c1,1 .. . cn−1,1 z
, n ≥ 1,
. . . cn−1,n
... zn ... ...
c0,n c1,n .. .
P0 = 1.
(2)
Define the infinite upper triangular matrix B = (bk,n )∞ k,n=0 with bk,n = 0 if k > n. This matrix can be considered as the infinite transition matrix from the algebraic basis n ∞ B = {Pn (z)}∞ n=0 to B = {z }n=0 . Indeed, each finite section Bn of B is the corresponding transition matrix from the algebraic basis Bn = {P0 (z), . . . , Pn (z)} in the linear vector space Pn [z] of polynomials of degree at most n, to the standard basis Bn = {1, z, . . . , z n }. Since Mn , In are both matricial representations of the same inner product with respect to Bn , Bn respectively, then Btn Mn Bn = In , t and consequently M−1 n = Bn Bn . This result in the case of Hankel matrices going back to A.C. Aitken, cf. [10] has been recovered several times see [5,8] (also [12] in the context of moment Hermitian matrices). In particular, if An = Bn Btn , it follows that
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Mn An = An Mn = In .
(3)
A natural question related to the above equality is to know if (3) remains true for infinite matrices in the particular case that the matrix multiplication A = BBt is well defined, that is, if it is true that A is the classical inverse of M in the sense that MA = AM = I.
(4)
The purpose of the present paper is to study this problem in the general framework of HPD matrices or in the particular case of Toeplitz matrices, and some questions related with it. In this direction in [8] the authors studied this problem in the particular case of positive definite Hankel matrices H which are moment matrices associated with measures supported in the real line. They stated that it is likely that the answer is positive in the indeterminate case or equivalently, as a consequence of [5], when limn→∞ λn > 0, where λn is the smallest eigenvalue of the finite section Hn of H. Our approach is also from the point of view of the asymptotic behaviour of the smallest eigenvalues λn of the finite sections of the HPD matrices. We prove that the answer is negative in the case of Toeplitz matrices, see Example 1. Our results do not contribute to the question about Hankel matrices. Recall that an HPD matrix M is the moment matrix associated with a Borel measure μ with support on the complex plane C if for all i, j ≥ 0 z i z j dμ.
ci,j =
In this case we denote M = M(μ). Note that the inner product (1) induced by M(μ) in P[z] is the inner product in L2 (μ), indeed, if p(z), q(z) ∈ P[z] p, q =
p(z)q(z)dμ.
For more information concerning the characterization of HPD matrices which are moment matrices with respect to a certain measure μ with support on C see among others [2,7,18]. Since moment matrices associated with measures with infinite support on C are HPD matrices, many of the examples that appear in this paper are indeed moment matrices. Note that in this particular case the corresponding orthonormal polynomials with respect to μ and having positive leading coefficients bn,n (μ) are given as Pn (z) =
n
bk,n (μ)z k .
k=0
In the context of the theory of orthogonal polynomials our aim is to study if we can compute the classical inverse of a moment matrix in terms of the coefficients of the orthonormal polynomials. For finite matrices the approach of determining the inverse of
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a finite Hankel or Toeplitz matrix in connection with the theory of orthogonal polynomials appears in [21] and [22] where an algorithm for the inversion of such matrices is obtained. Later on, this point of view is also treated in [16] for certain finite moment Hankel matrices; in the finite dimensional case this kind of algorithms enables to compute inverse matrices faster. We point out that our results are for infinite dimensional Hermitian matrices. The paper is organized as follows: in Section 2 we provide a counterexample showing that the matrix A is not, in general, a classical inverse of an HPD matrix M when limn→∞ λn > 0. However, our example is not a Hankel matrix, and in this case the problem remains open. We show that in the particular case of HPD matrices with square summable rows the condition limn→∞ λn > 0 is sufficient to assure that A is a classical inverse of M. This condition is not a necessary condition as we show. Section 3 is devoted to study the main problem in the context of HPD Toeplitz matrices. In this direction, using the persymmetric property of the inverse of a finite Toeplitz matrix we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the classical inverse of an HPD Toeplitz matrix is an asymptotic Toeplitz matrix. Moreover, since every HPD Toeplitz matrix is a moment matrix associated with a measure on the unit circle T (see e.g. [1]), we give the description of the classical inverse of an HPD Toeplitz matrix in terms of the limits of the coefficients of the orthonormal polynomials. We apply our techniques to the inversion of Toeplitz operators not necessarily bounded. As a consequence we obtain that the asymptotic limit of the inverse of a Toeplitz matrix Tϕ associated with a continuous symbol verifying inf z∈T ϕ(z) > 0 is the Toeplitz matrix T ϕ1 . Finally, in Section 3 we also obtain some applications of the preceding results for orthonormal polynomials. In particular, we show that whenever limn→∞ λn (ν) > 0 we may assure the existence of all the limits of the coefficients of the orthonormal polynomials limn→∞ bn−k,n (ν) for every k ≥ 0. Note that in the case of the main coefficients the existence of limn→∞ bn,n (ν) was already known by Szegö theory; the case of the other coefficients, up to our knowledge, was not known. First, we introduce some notation. For A = (ai,j )ni,j=0 being a finite matrix we identify the linear operator on Cn+1 induced by A with its matrix with respect to the standard basis of Cn+1 . However in the infinite case we distinguish between infinite matrices and operators using different ∞ notation. If A is an infinite matrix and the matrix product ∞ t Ax = exists for every x ∈ 2 and belongs to 2 , then the graph is j=0 ai,j xj i=0 closed and by the closed graph theorem A defines a bounded operator that we will denote by A. Note that there are infinite matrices not defining bounded operators, consider for t example the diagonal matrix A = (iδi,j )∞ i,j=0 , since the product Ax exists for all x ∈ 2 but the result does not always belong to 2 . In [11] a criterion has been proved to characterize when an infinite matrix defines a bounded operator from 2 to 2 . On the other hand, it is well known that if A : H → H is a bounded operator on a countable Hilbert space H with an orthonormal basis B = {vi }∞ i=0 then the associated matrix is A = (Avj , vi )∞ . i,j=0
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2. Infinite transition matrices. Inversion of certain HPD matrices Let M be an HPD matrix and let B be the transition matrix associated with M. t Let A = (ai,j )∞ such formal matrix product is i,j=0 be the matrix A = BB whenever ∞ well defined, i.e. if for all i, j ∈ N0 there exists ai,j = k=max{i,j} bi,k bj,k . In the case of M = M(μ) being a moment matrix associated with a measure μ we denote by A = A(μ). Remark 1. Note that the existence of A = BBt is equivalent to all the rows of the matrix B being square summable. One can ask if the matrix B comes from a bounded operator in a Hilbert space with a countable orthonormal basis like 2 or the completion of the polynomials with the inner product from M. In a paper of Berg–Durán [6] the authors consider an operator T on the polynomials given by T (z n ) = Pn . Since T (Pn ) =
n
bk,n Pk
k=0
the operator T has the matrix B with respect to the orthonormal basis {Pn (z)}∞ n=0 . Then Theorem 1.1 in [6] can be reformulated as: Lemma 1. Let M be an infinite HPD matrix and let B be the transition matrix associated with M. Then, the following are equivalent: (1) limn→∞ λn = λ > 0. (2) The matrix B defines a bounded operators B on 2 and B = λ−1/2 . As a consequence of (1) we give a sufficient condition in terms of the asymptotic behaviour of λn to assure the existence of the matrix A defining a bounded operator A on 2 . Lemma 2. Let M be an HPD matrix such that limn→∞ λn = λ > 0, then the matrix ∗ A = BBt = (ai,j )∞ i,j=0 exists and defines the bounded operator A = BB on 2 . Moreover, (1) ai,j = limn→∞ M−1 n [i, j] for every i, j ∈ N0 . (2) A = B2 = λ−1 . Proof. By Lemma 1 the matrices B, Bt define bounded operators on 2 and consequently the rows and columns of such matrices belong to 2 . Then for every i, j ∈ N0 the series ∞ k=max{i,j} bi,k bj,k is absolutely convergent and ai,j =
∞
bi,k bj,k = lim
n
n→∞
k=max{i,j}
bi,k bj,k = lim M−1 n [i, j]. n→∞
k=max{i,j}
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Therefore the matrix A exists and is the representation with respect to the standard ∗ ∗ basis of 2 of the operator BB . Moreover,A = BB 2 = B2 = λ−1 . 2 Remark 2. In the case limn→∞ λn = 0 we cannot assure even the existence of the matrix A, as the following example shows: consider the Lebesgue measure μ on the circle with center (1, 0) and radius 1, and the associated moment matrix ⎛
1 ⎜1 ⎜ ⎜ 1 M(μ) = ⎜ ⎜ ⎜1 ⎝ .. .
1 1 1 2 3 4 3 6 10 4 10 20 .. .. .. . . .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .
The sequence of orthonormal polynomials associated with μ is Pn (z) = (z − 1)n for all ∞ n ≥ 0 and since k=0 |b0,k (μ)|2 = ∞ it follows that A(μ) does not exist. Note that n−k n the transition matrix is B(μ) = (bk,n (μ))∞ k,n=0 with bk,n (μ) = (−1) k if k ≤ n and bk,n = 0 if k > n which obviously does not define a bounded operator on 2 . We show that the answer of the main problem is negative for HPD matrices: Example 1. There exists a Toeplitz HPD matrix T such that limn→∞ λn > 0 and the matrix A is not a classical inverse of T, i.e., TA = I,
AT = I.
Indeed, consider the moment matrix T(ν) associated with the measure ν = 12 m + μ0 where m is the Lebesgue measure on T and μ0 is the atomic measure supported on the point 1 and with μ0 ({1}) = 12 , that is ⎛
1
⎜ 12 ⎜1 ⎜ T(ν) = ⎜ 2 ⎜1 ⎝2 .. .
1 2
1 1 2 1 2
.. .
1 2 1 2
1 1 2
.. .
1 2 1 2 1 2
1 .. .
⎞ ... ...⎟ ⎟ ...⎟ ⎟. ...⎟ ⎠ .. .
Since T(ν) ≥ 12 I in the sense that for every v ∈ c00 we have vT(ν)v ∗ ≥ 12 vv ∗ , it follows that λn (ν) ≥ 12 for all n ∈ N0 and limn→∞ λn (ν) ≥ 12 . It can be checked that 2(n+1) 2 −1 M−1 n [i, j] = − n+2 if i = j and Mn [i, i] = n+2 for each n ∈ N0 and 0 ≤ i, j ≤ n. By using Lemma 2 it follows that ai,j = limn→∞ M−1 n [i, j] = 2δi,j and A(ν) = 2I. Therefore A(μ)T(ν) = I
and
T(ν)A(ν) = I.
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Using the efficient numerical algorithms of Cholesky decomposition and inversion of lower triangular matrices we can determine the explicit representation of the transition matrix √ √ √ 2n and bk,n = − 2 B(ν) = (bk,n )∞ with b = if k < n. n,n k,n=0 n+1 n(n+1) In the following proposition we prove that the condition limn→∞ λn > 0 is a sufficient condition for A being the inverse of an HPD matrix M in the particular case of M having square summable rows. Remark 3. Let M = (ci,j )∞ i,j=0 be an HPD matrix with square summable rows, i.e. ∞ 2 j=0 |ci,j | < ∞ for all i ≥ 0; then the columns are also square summable. Consider c00 the subspace of 2 that consists of all complex sequences with only finitely many non-zero entries. It is clear that Metk ∈ 2 being {ek } the standard basis of 2 , consequently Mxt ∈ 2 for every x ∈ c00 . Therefore the linear mapping M : c00 → 2 given by the formal matrix product Mxt is well defined, not necessarily defining a bounded operator. Proposition 1. Let M = (ci,j )∞ i,j=0 be an HPD matrix with square summable rows. Assume that limn→∞ λn > 0 then AM = MA = I. Proof. First of all since M is Hermitian it has square summable rows and columns. By Lemma 2 since limn→∞ λn > 0 it follows that the matrix A is the matrix representation of a bounded operator on 2 and, consequently, the rows and columns of A belong to 2 . Then both matrices AM and MA are well defined. We first show that for each j, k ∈ N0 AM[j, k] = δj,k . We introduce the following notation: for every n ≥ 0 denote by Πn the projection Πn : 2 → 2 defined by Πn (x0 , x1 , . . . , xn , . . .) = (x0 , x1 , . . . , xn , 0, 0, . . .) for every x ∈ 2 . Let B be a bounded on 2 , we denote by B˜n = Πn BΠn , which matrix representation operator Bn 0 is given by . It is well known (see e.g. [14]) that {B˜n }∞ n=0 is strongly convergent 0 0 ˜ }∞ and {B˜∗ }∞ are strongly convergent to B, B∗ respectively, to B on 2 . Since {B n n=0 n n=0 ∗ ˜∗ ∞ ˜ then {Bn Bn }n=0 is strongly convergent to BB . Fix k ∈ N0 , since (ci,k )∞ i=0 ∈ 2 it follows that for every n ≥ k we have that ⎛ AMetk
= lim
n→∞
Bn B∗n 0
c0,k c1,k .. . cn,k
⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ Bn B∗n ⎟ ⎜ ⎟ = lim n→∞ ⎜ ⎟ 0 0 ⎜ ⎟ ⎜ cn+1,k ⎟ ⎝ ⎠ .. .
⎛
⎞ c0,k ⎜c ⎟ ⎜ 1,k ⎟ ⎜ . ⎟ . ⎟ 0 ⎜ ⎜ . ⎟ ⎜ ⎟ = ek . 0 ⎜ cn,k ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎝ ⎠ .. .
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Therefore ej AMetk = δj,k . On the other hand, since M and A are Hermitian matrices MA = I.
2
Remark 4. Unfortunately Proposition 1 does not provide information in the context of Hankel matrices since if limn→∞ λn > 0 the Hankel matrix H = (si+j )∞ i,j=0 does not ∞ have square summable rows. Indeed, if n=0 s2n < ∞ it follows limn→∞ sn = 0. On the other hand, for every n ∈ N0 , s2n = e2n He∗2n ≥ λn and therefore limn→∞ λn = 0. Remark 5. The condition limn→∞ λn > 0 is not necessary for A being the classical inverse of an HPD matrix with square summable rows, even in the case of moment matrices as ∞ the following example shows: let 0 < a < 1 and M = amax{i,j} i,j=0 , that is ⎛
1 ⎜ a ⎜ ⎜ 2 a M=⎜ ⎜ 3 ⎜a ⎝ .. .
a a a2 a3 .. .
a2 a2 a2 a3 .. .
a3 a3 a3 a3 .. .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. . n(n+1)
It can be easily checked that M is positive definite since |Mn | = a 2 (1 − a)n > 0 ∞ for each n ∈ N0 . Consider the diagonal matrix D = ai/2 δi,j i,j=0 and the matrix |i−j| ∞ 1 Pr (θ) T= a 2 which is the Toeplitz matrix (see [17]) for the Poisson kernel 2π i,j=0 √ with r = a and Pr (θ) =
∞
r|n| einθ .
n=−∞
Then it holds that M = Dt TD. Taking in account this equality it is obvious that T is an HPD Toeplitz matrix and consequently T = T(ν) for a certain measure ν with support on T. Using [13] and [15] it follows that M is the moment matrix of the image √ measure μ = ν ◦ f −1 under the transformation f (z) = az. In this particular case it is easy to obtain the orthonormal polynomials by (2)
1
a 1
. Pn (z) = |Mn−1 ||Mn |
..
1 Consequently,
a . . . an
a . . . an z n−1 (z − a) , n = 1, 2, . . . .. ..
= n a (1 − a) . .
z . . . zn
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⎛
1
⎜ ⎜0 ⎜ ⎜ 0 B(μ) = ⎜ ⎜ ⎜0 ⎜ ⎝ .. .
−a a(1−a) 1 a(1−a)
0
0
0
−a a2 (1−a) 1 a2 (1−a)
0 .. .
0 .. .
...
0 −a a3 (1−a) 1 a3 (1−a)
.. .
⎞
⎟ ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎟ ⎠ .. .
In particular, B(μ) does not define a bounded operator on 2 and by Lemma 1 we have that limn→∞ λn (μ) = 0. On the other hand it can be checked that A(μ)M(μ) = M(μ)A(μ) = I, where ⎛
1 ⎜ −1 ⎜ 1 ⎜ ⎜ 0 A(μ) = 1−a⎜ ⎜ 0 ⎝ .. .
−1 a+1 a −1 a
0 .. .
0 −1 a a+1 a2 −1 a2
.. .
0 0 −1 a2 a+1 a3
.. .
⎞ ... ...⎟ ⎟ ⎟ ...⎟. ⎟ ...⎟ ⎠ .. .
3. Inversion of Toeplitz HPD matrices It is well known that the inverse of a Toeplitz matrix is not, in general, a Toeplitz matrix, even in the finite dimensional case. Nevertheless, such inverse is persymmetric as explained below. For an (n + 1) × (n + 1) matrix A = (ai,j )ni,j=0 introduce the matrix Aπ by Aπ [i, j] = A[n − j, n − i] = an−j,n−i ,
i, j = 0, . . . , n
Then Aπ = JAt J, where J[k, l] = 1 if and only if k + l = n and otherwise J[k, l] = 0 (i.e. zeros except in the south-west to north-east diagonal having ones). Clearly (AB)π = Bπ Aπ , so if A is regular then so is Aπ , and in the affirmative case (Aπ )−1 = (A−1 )π . The matrix A is called persymmetric iff A = Aπ , i.e., if and only if it is symmetric with respect to the south-west to north-east diagonal. A Toeplitz matrix T is clearly persymmetric. Therefore if A is regular and persymmetric then A−1 is persymmetric. In particular, if T is a regular Toeplitz matrix, then T−1 is persymmetric. We show that this property has a great impact on the form of a classical inverse of certain infinite Toeplitz matrices. In order to do it we introduce the notion of asymptotic Toeplitz matrix very closely related to the notion of asymptotic Toeplitz operator that appears in [4]; indeed, in the particular case of matrices defining bounded operators in
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Hilbert spaces, these matrices are the representations of such operators with respect to orthonormal basis. Definition 1. An infinite matrix A = (ai,j )∞ i,j=0 is an asymptotic Toeplitz matrix if for every k ∈ Z there exists limn→∞ an,n+k = αk . In such a case we denote by Lim(A) = (αi−j )∞ i,j=0 which is clearly a Toeplitz matrix. The main result in this section is: Theorem 1. Let T be an infinite HPD Toeplitz matrix and let B = (bk,n )∞ k,n=0 be the transition matrix associated with T. Assume that the matrix A = BBt = (ai,j )∞ i,j=0 exists. Then: (1) B is an asymptotic Toeplitz matrix, i.e. for every k limn→∞ bn−k,n = βk . (2) For i, k ∈ N0 with i ≤ k, ai,k =
i
∈
N0 there exists
βj βk−j .
j=0
(3) Moreover, if
∞ k=0
βk2 < ∞ then the matrix A is an asymptotic Toeplitz and lim ak,k+j =
k→∞
∞
βi βi+j ,
i=0
where the series above is absolutely convergent. Proof. We first show that there exists limn→∞ bn,n . Indeed, since A exists and T−1 n = Bn Btn a0,0 =
∞ k=0
|b0,k |2 = lim
n→∞
n k=0
|b0,k |2 = lim T−1 n [0, 0]. n→∞
−1 Using that the matrix Tn−1 is persymmetric it follows that T−1 n [0, 0] = Tn [n, n] and therefore 2 a0,0 = lim T−1 n [n, n] = lim bn,n . n→∞
n→∞
√ Moreover β0 = limn→∞ bn,n = a0,0 > 0 since bn,n > 0 for each n ∈ N0 . Now we prove the existence of limn→∞ bn−k,n for every k ∈ N. Indeed, using again that T−1 n is persymmetric −1 a0,k = lim T−1 n [0, k] = lim Tn [n − k, n] = lim bn−k,n bn,n . n→∞
n→∞
n→∞
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Since limn→∞ bn,n = β0 > 0 we have that there exists limn→∞ bn−k,n and moreover, lim bn−k,n =
n→∞
a0,k . β0
In a general way we can determinate all of the entries of the matrix A. Indeed, let i, k ∈ N0 be fixed with i ≤ k then −1 t T−1 n [i, k] = Tn [n − k, n − i] = (Bn Bn )[i, k] =
i
bn−i,n−i+j bn−k,n−i+j .
j=0
By passing to the limit and using Lemma 2 ⎛
ai,k = lim T−1 n [i, k] = n→∞
i
⎞ βk−i ⎟ ⎜ ⎜ ... ⎟ ⎜ ⎟. · · · βi ⎜ ⎟ ⎝ βk−1 ⎠ βk
β j βk+j−i =
β0
β1
j=0
Thus, we have the description of A. Note that the entries of the main diagonal of A are k given by ak,k = i=0 βi2 . In order to prove (3) let j ∈ N0 be fixed and k ∈ N, then ⎛
ak,k+j =
The series exists
∞ i=0
β0
β1
⎞ βk+j−k ⎟ k .. ⎜ ⎜ ⎟ . ⎜ ⎟= βi βi+j . · · · βk ⎜ ⎟ ⎝ βk−1 ⎠ i=0 βk+j
βi βi+j is absolutely convergent since
lim ak,k+j =
k→∞
∞
β i βi+j .
∞ k=0
βk2 < ∞, and therefore there
2
i=0
Remark 6. Note that in Example 1 the corresponding matrix B(ν) associated with ν is √ asymptotic Toeplitz and limn→∞ bn−k,n (ν) = 0 if k > 0 and limn→∞ bn,n (ν) = 2. As a consequence of the results of the preceding section we obtain the explicit representation of the inverse of an HPD Toeplitz matrix with limn→∞ λn > 0 in terms of the coefficients of the orthonormal polynomials. The existence of the classical inverse in this case was known (see e.g. [9]) in the context of the nonhomogeneous solutions of systems of infinite equations.
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Corollary 1. Let T = (ci−j )∞ i,j=0 be an HPD Toeplitz matrix with limn→∞ λn > 0 and ∞ 2 j=0 |cj | < ∞. Then the matrix A is the classical inverse of T and verifies that for i, k ∈ N0 with i ≤ k, i
ai,k =
βj βk−j ,
j=0
where βk = limn→∞ bn−k,n for k ≥ 0. The main consequence of Theorem 1 in the context of the orthogonal polynomials is that, in the case of limn→∞ λn > 0, we may assure the existence of the limits of the coefficients of the orthonormal polynomials associated with a measure with support on T. Corollary 2. Let ν be a measure with in infinite support on T and let Pn (z) = n k the orthonormal polynomial with respect to ν. Assume that k=0 bk,n (ν)z limn→∞ λn > 0, then for every k ∈ N0 there exists (is finite) the following limit: lim bn−k,n (ν).
n→∞
Moreover, 0 < limn→∞ bn,n (ν) < ∞. Remark 7. We may rewrite the result above to obtain a consequence in the asymptotic (1) (2) ∞ behaviour of the connection coefficients. Recall that if {Qn }∞ n=0 and {Qn }n=0 are two system of monic polynomials corresponding to the measures μ1 and μ2 and Q(2) n (z) =
n
c(k, n)Q(1) n (z),
k=0 (1)
(2)
we call {c(k, n)}k,n the connection coefficients from {Qn } to {Qn } (see e.g. [20]). In particular, if ν is a measure on T and m is the Lebesgue measure, since {z n }∞ n=0 is the sequence of monic polynomials with respect to m we may rewrite the system of monic polynomials with respect to ν as: Qn (z) =
n
c(k, n, ν)z k
k=0
bk,n (ν) and c(n, n, ν) = 1. As a consequence of the results in the bn,n (ν) preceding section we have: where c(k, n, ν) =
Corollary 3. Let ν be a measure with infinite support on T and let {Qn (z)}∞ n=0 be the corresponding sequence of monic orthogonal polynomials given by:
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Qn (z) =
n
c(k, n, ν)z k .
k=0
If limn→∞ λn (ν) > 0, then for every k ∈ N0 there exist (are finite) lim c(n, n + k, ν)
n→∞
Proof. Note that if limn→∞ λn (ν) > 0 then the matrix A(ν) exists and therefore by Theorem 1 the matrix B(ν) is asymptotic Toeplitz, and moreover lim bn,n (ν) = κ(ν) > 0
n→∞
Then, we may conclude that for every k ∈ N0 there exists (is finite) the following limit: bn,n+k (ν) . n→∞ bn,n (ν)
lim c(n, n + k, ν) = lim
n→∞
2
We finish this section with some consequences of the inversion result of Toeplitz matrices defining, in general, not necessarily bounded Toeplitz operators. Corollary 4. Let Tϕ = (ci−j )∞ i,j=0 be a Toeplitz HPD matrix associated with a symbol ϕ ∈ L2 (T) and ess inf ϕ(z) > 0. Then the matrix Aϕ = BBt exists and is the classical inverse matrix of Tϕ , i.e. Aϕ Tϕ = Tϕ Aϕ = I. In the particular case of bounded Toeplitz matrices we obtain: Proposition 2. Let Tϕ be an HPD Toeplitz matrix with continuous symbol ϕ such that inf ϕ(z) > 0 then Aϕ is asymptotic Toeplitz and Lim(Aϕ ) = T ϕ1 . 1 ϕ ϕ, ϕ1
Proof. Since over, since
is continuous on T then T ϕ1 defines a bounded Toeplitz operator. More∈ L∞ (T) by [4] it follows that
I − Tϕ T ϕ1 = Hzϕ Hz ϕ1 = K Now, by [3] since the symbols zϕ and z ϕ1 are continuous then the Hankel matrices Hzϕ , Hz ϕ1 define compact operators and consequently K defines a compact operator. Therefore K is an asymptotic matrix with Lim(K) = 0. Since A defines a bounded operator which is the inverse operator of Tϕ then Tϕ (A − T ϕ1 ) = Tϕ A − Tϕ T ϕ1 = K, and A − T ϕ1 = ATϕ (A − T ϕ1 ) = AK.
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Since A defines a bounded operator and K defines a compact operator then the matrix AK is the matrix representation of a compact operator and therefore such matrix is weakly asymptotically Toeplitz with limit 0 and we have the conclusion. 2 Acknowledgements This work has been partially supported by the project TEC2012-35673, funded by Ministerio de Economía y Competitividad, Spain. References [1] N.I. Akhiezer, The Classical Moment Problem, Oliver and Boyd Ltd., Edinburgh and London, 1965. [2] A. Atzmon, A moment problem for positive measures on the unit disc, Pacific J. Math. 59 (1975) 317–325. [3] R.A. Martinez-Avendaño, P. Rosenthal, An Introduction to Operators on the Hardy–Hilbert Space, Graduate Texts in Mathematics, vol. 237, 2007. [4] J. Barriá, P.R. Halmos, Asymptotic Toeplitz operators, Trans. Amer. Math. Soc. 273 (1982) 621–630. [5] C. Berg, Y. Chen, M.E.H. Ismail, Small eigenvalues of large Hankel matrices: the indeterminate case, Math. Scand. 91 (2002) 67–81. [6] C. Berg, A.J. Durán, Orthogonal polynomials and analytic functions associated to positive definite matrices, J. Math. Anal. Appl. 315 (2006) 54–67. [7] C. Berg, P. Maserick, Exponentially bounded positive definite functions, Illinois J. Math. 28 (1984) 162–179. [8] C. Berg, R. Szwarc, The smallest eigenvalue of Hankel matrices, Constr. Approx. 34 (2011) 107–133. [9] M. Bernkopf, A history of infinite matrices, Arch. Hist. Exact Sci. 4 (1968) 308–358. [10] A.R. Collar, On the reciprocation of certain matrices, Proc. R. Soc. Edinb. (1939) 301–312. [11] L. Crone, A characterization of matrix operators on 2 , Math. Z. 123 (1991) 315–317. [12] C. Escribano, R. Gonzalo, E. Torrano, Small eigenvalues of large Hermitian moment matrices, J. Math. Anal. Appl. (2010). [13] C. Escribano, M.A. Sastre, E. Torrano, Moment matrix of self-similar measures, Electron. Trans. Numer. Anal. 24 (2006) 79–87. [14] P.R. Halmos, A Hilbert Space Problem Book, second edition, Springer-Verlag, New York, 1980. [15] P.E.T. Jorgensen, K.A. Kornelson, K.L. Shuman, Iterated Function Systems, Moments, and Transformations of Infinite Matrices, Memoirs of the American Mathematical Society, vol. 213 (1003), 2011. [16] S.B. Provost, H. Ha, On the inversion of certain moment matrices, Linear Algebra Appl. 430 (2009) 2650–2658. [17] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987. [18] F.H. Szafraniec, Boundedness of the shift operator related to definite positive forms: an application to moment problem, Ark. Math. 19 (1981) 251–259. [19] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 32, 1939. [20] R. Szwarc, Connection coefficients of orthogonal polynomials, Canad. Math. Bull. 35 (4) (1992) 548–856. [21] W.F. Trench, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Ind. Appl. Math. 12 (1964) 515–522. [22] W.F. Trench, An algorithm for the inversion of finite Hankel matrices, J. Soc. Ind. Appl. Math. 13 (4) (1965) 1102–1107.