Dyukarev–Stieltjes parameters of the truncated Hausdorff matrix moment problem

Abstract

We obtain a new multiplicative decomposition of the resolvent matrix of the non-degenerate truncated Hausdorff matrix moment (THMM) problem in the case of odd and even number of moments with the help of Dyukarev–Stieltjes matrix parameters (DSMP). Our result generalizes the Dyukarev representation of the resolvent matrix of the truncated Stieltjes matrix moment problem published in (Math Notes 75(1–2):66–82, 2004). In the scalar case, these parameters appear in the celebrated Stieltjes’s (1894) work Recherches sur les fractions continues and are used to establish the determinateness of the moment problem. We also obtain explicit relations between four families of orthogonal matrix polynomials on [a, b] together with their matrix polynomials of the second kind and the DSMP of the THMM problem. Additionally, we derive new representations of the Christoffel–Darboux kernel.

Appendices

Appendix 1: Orthogonal matrix polynomials on [a, b]

Let us reproduce some notions on OMP which were introduced in [10]. Let P be a complex $p \times q$ matrix polynomial. For all $n\in \mathbb N_0$, let

$$\begin{aligned} Z^{[P]}_{n}:=[A_0,A_1,\cdots ,A_n], \end{aligned}$$

where $(A_j)_{j=0}^\infty$ is the unique sequence of complex $p \times q$ matrices such that for all $z\in \mathbb C$ the polynomial P admits the representation $P(z)=\sum _{j=0}^\infty z^jA_j$.

Furthermore, we denote by $\mathrm{deg}\, P:=\sup \{j\in \mathbb N_0: A_j\ne 0_{p\times q}\}$ the degree of P. Observe that in the case $P(z)=0_{p\times q}$ for all $z\in \mathbb C$ we have thus $\mathrm{deg}\, P=-\infty$. If $k:=\mathrm{deg}\, P\ge 0$, we refer to $A_k$ as the leading coefficient of P.

For all $k\in \mathbb N_0$ and all $\kappa \in \mathbb N_0$ with $k\le \kappa$, let ${\mathbb Z}_{k,\kappa }:=\{n\in \mathbb N_0, k\le n\le \kappa \}$.

Definition 8

Let $\kappa \in \mathbb N_0\cup \{\infty \}$, and let $(s_j)_{j=0}^{2\kappa }$ be a sequence of complex $q\times q$ matrices. A sequence $(P_k)_{k=0}^\kappa$ of complex $q\times q$ matrix polynomials is called a monic left orthogonal system of matrix polynomials with respect to $(s_j)_{j=0}^{2\kappa }$ if the following three conditions are fulfilled.

• (I) $\deg P_k =k$ for all $k\in {\mathbb Z}_{0,\kappa }$.

• (II) $P_k$ has the leading coefficient $I_q$ for all $k\in {\mathbb Z}_{0,\kappa }$.

• (III) $Z_n^{[P_j]}H_n(Z_n^{[P_k]})^*= 0_{q\times q}$ for all $j,k\in {\mathbb Z}_{0,\kappa }$ with $j\ne k$, where $n:=\max \{j,k\}$.

Remark 7

[10, Remark 3.6] Let $n\in \mathbb N_0\cup \{\infty \}$, and let $(s_j)_{j=0}^{2n}$ be a Hausdorff positive definite sequence, i.e., the corresponding Hankel block matrix $H_{n}$ is positive definite. Denote by $(P_k)_{k=0}^n$ the monic left orthogonal system of matrix polynomials with respect to $(s_j)_{j=0}^{2n}$. Let $\sigma$ be a nonnegative Hermitian $q\times q$ measure on $\mathbb R$ satisfying $s_j=\int _{[a,b]}t^j \mathrm{d}\sigma (t)$ for $0\le j\le 2n$. Thus,

$$\begin{aligned} \int _{[a,b]}P_j\mathrm{d}\sigma P_k^*=\left\{ \begin{array}{cc} \widehat{H}_j, &{}\quad \text{ if }\ \ j=k, \\ 0_q, &{}\quad \text{ if }\ \ j\ne k\\ \end{array} \right. \end{aligned}$$

for all $0\le j,k\le n$ where $\widehat{H}_j$ denotes the Schur complement of $H_{j-1}$ in $H_j$; see (1.9).

Definition 9

Let $(s_k)_{k=0}^{2j}$ be a Hausdorff positive definite sequence. Let

$$\begin{aligned} P_{1,0}(z)&:= I_q, \ \ P_{2,0}(z):=I_q, \ Q_{1,0}(z):=0_{q}, \ Q_{2,0}(z,a,b):=-(u_{2,0}+z\,s_0),\\ P_{1,j}(z)&:=(-Y^{*}_{1,j}H^{-1}_{1,j-1},I_q)R_{j}(z)v_{j}, \quad 1\le j\le n,\\ P_{2,j}(z,a,b)&:=(-Y^{*}_{2,j}H^{-1}_{2,j-1},I_q)R_{j}(z)v_{j},\quad 1\le j\le n-1,\\ Q_{1,j}(z)&:=-(-Y^{*}_{1,j}H^{-1}_{1,j-1},I_q)R_{1,j}(z) u_{1,j},\quad 1\le j\le n, \end{aligned}$$

and

$$\begin{aligned} Q_{2,j}(z,a,b):=-(-Y^{*}_{2,j}H^{-1}_{2,j-1},I_q)R_{j}(z) (u_{2,j}+zv_{j}s_{0}), \quad 1\le j\le n-1. \end{aligned}$$

Definition 10

Let $K_{k,j}$, $\widetilde{u}_{k,j}$, $\widetilde{Y}_{k,j}$, for $k=1,2$, $R_j$ and $v_j$ be as in (1.5), (1.6), (1.21), (1.22), (1.23), (1.7) and (1.8), respectively.

Let $(s_k)_{k=0}^{2j+1}$ be a Hausdorff positive definite sequence. Let

$$\begin{aligned} \Gamma _{1,0}(z)&:=I_q, \, \Gamma _{2,0}(z):=I_q, \ \ \Theta _{1,0}(z):=s_0,\, \Theta _{2,0}(z):=-s_0 \end{aligned}$$

for all $z\in \mathbb C$. For $k\in \{1,2\}$ and $1\le j\le n$, define

$$\begin{aligned} \Gamma _{1,j}(z,b)&:=(-\widetilde{Y}^{*}_{1,j}K^{-1}_{1,j-1},I_q)R_{j}(z)v_{j},\nonumber \\ \Gamma _{2,j}(z,a)&:=(-\widetilde{Y}^{*}_{2,j}K^{-1}_{2,j-1},I_q)R_{j}(z)v_{j}, \end{aligned}$$

(4.21)

$$\begin{aligned} \Theta _{1,j}(z,b)&:= (-\widetilde{Y}^{*}_{1,j}K^{-1}_{1,j-1},I_q)R_{j}(z)\widetilde{u}_{1,j},\nonumber \\ \Theta _{2,j}(z,a)&:= (-\widetilde{Y}^{*}_{2,j}K^{-1}_{2,j-1},I_q)R_{j}(z)\widetilde{u}_{2,j} \end{aligned}$$

(4.22)

for all $z\in \mathbb C$.

We usually omit the dependence of the polynomials $P_{k,j}$, $Q_{k,j}$, $\Gamma _{k,j}$ and $\Theta _{k,j}$ for $k=1,2$ on the parameters a and b.

In [3], (resp. [34]) it was proved that polynomials $P_{k,j}$ (resp. $\Gamma _{k,j}$) for $k=1,2$ are in fact OMP on [a, b]. In [4] some properties of second kind polynomials $Q_{k,j}$ and $\Theta _{k,j}$ for $k=1,2$ were discussed. In [10] explicit interrelations between $P_{k,j}$, $\Gamma _{k,j}$ and their second kind polynomials were studied.

For the sake of completeness in the following remark, we reproduce explicit interrelations between the matrices $\widehat{H}_{k,j}$, $\widehat{K}_{k,j}$ and the polynomials $P_{1,j}$, $Q_{2,j}$, $\Gamma _{1,j}$, $\Theta _{2,j}$ considered in Corollary 3.4 and Corollary 3.10 in [4].

Remark 8

Let $\widehat{H}_{k,j}$, $\widehat{K}_{k,j}$, for $k=1,2$, $P_{1,j}$, $Q_{2,j}$, $\Gamma _{1,j}$ and $\Theta _{2,j}$ be as in (1.9), (1.10), (1.24), (1.25) and Definitions 9 and 10, respectively. The following equalities then hold

$$\begin{aligned} \widehat{H}_{1,j}=&-P_{1,j}(a)\Theta _{2,j}^*(a),\nonumber \\ \widehat{H}_{2,j}=&-Q_{2,j}(a)\Gamma _{1,j+1}^{*}(a), \end{aligned}$$

(4.23)

$$\begin{aligned} \widehat{K}_{1,j}&=\Gamma _{1,j}(a)Q_{2,j}^*(a), \nonumber \\ \widehat{K}_{2,j}&=\Theta _{2,j}(a)P_{1,j+1}^{*}(a). \end{aligned}$$

(4.24)

Finally, let us recall the following well-known result below.

Lemma 1

[1, Proposition 8.2.4] Let $A:=\left( \begin{array}{cc} A_{11} &{} A_{12} \\ A_{12}^* &{} A_{22} \end{array} \right)$ be a Hermitian $(n+m)\times (n+m)$ matrix. Therefore, the following statements are equivalent:

• (i) $A>0$.

• (ii) $A_{11}>0$ and $A_{12}^*A_{11}^{-1}A_{12}<A_{22}$.

• (iii) $A_{22}>0$ and $A_{12}A_{22}^{-1}A_{12}^*<A_{11}$.

Appendix 2: The Christoffel–Darboux kernel

In this section, we indicate a new form of the Christoffel–Darboux (CD) kernel in terms of the polynomials $\Gamma _{2,j}$ and $\Theta _{2,j}$. We also consider the relation between the CD kernel and the DSMP. Additionally, we also introduce new CD type kernels.

Let us recall the three-term recurrence relation for the OMP $P_{1,j}$ via the Schur complements $\widehat{H}_{1,j}$; see [7, Definition 8.1].

Definition 11

Let $(s_k)_{k=0}^{\infty }$ be a positive definite sequence, i.e., the block matrices $H_{1,j}$ for $j\in \mathbb N_0$ are positive definite. Let $\widetilde{H}_{2,j}$ be as in (1.4) and $\widehat{H}_{1,j}$ be defined by (1.9). Define $A_{0}:=s_1s_0^{-1}$, $B_{-1}:=0_q$, $B_{0}:=\widehat{H}_{1,0}^{-1}\widehat{H}_{1,1}$. For $1\le j\le n$, let $A_{j}:=E_{j}\widehat{H}_{1,j}^{-1}$, $B_{j}:=\widehat{H}_{1,j}^{-1}\widehat{H}_{1,j+1}$, with $E_{j}:=(-Y_{1,j}^*H_{1,j-1}^{-1},\,I)\widetilde{H}_{2,j} \left( \begin{array}{c} -H_{1,j-1}^{-1}Y_{1,j} \\ I_q \end{array} \right)$. The ordered pair $[(A_k)_{k=0}^\infty ,(B_k)_{k=0}^\infty ]$ is called the Favard pair (abbreviated as F-pair) associated with $(s_k)_{j=0}^{\infty }$.

The following proposition is proven in [10, Proposition 3.7]; see also [7, Remark 8.7].

Proposition 8

The polynomials $P_{1,j}$ satisfy the recurrence relations

$$\begin{aligned} x P_{1,j}(x)=B_{j-1}^*P_{1,j-1}(x)+A_{j}P_{1,j}(x)+P_{1,j+1}(x),\quad j\ge 0, \end{aligned}$$

with initial conditions $P_{1,-1}(x)=0_q$ and $P_{1,0}(x)=I_q$.

Now we turn to the notion of the matrix CD kernel, which is defined for $z,\, w\in \mathbb C$ by

$$\begin{aligned} K_n(z,w):=\sum _{j=0}^n P_{1,j}^*(\bar{z})\widehat{H}_{1,j}^{-1}P_{1,j}(w), \end{aligned}$$

where $P_{1,j}$, according to [11, Formula (2.22) and (2.23)], are left OMP. In the Damanik et al. [11] work, the orthonormal matrix polynomial $\widehat{H}_{1,j}^{-\frac{1}{2}}P_{1,j}$ instead of the monic OMP $P_{1,j}$ is employed.

Different representations of the matrix kernel $K_n$ were studied in [11, p. 34], [21, Theorem 2.6], [14, Formula (3.29)], [13, Lemma 2.1] and by other authors.

Proposition 9

Let a be real number. Let $H_{1,n}$, $K_{2,n}$ defined by (1.2) and (1.6) be positive definite block matrices. Let $R_n$, $v_n$, $\widehat{H}_{1,n}$, $P_{1,n}$, $\Gamma _{2,n}$ and $\Theta _{2,n}$ be as in (1.7), (1.8), (1.9), as well as Definitions 9 and 10. Therefore, the following equalities hold for $z\in \mathbb C$:

$$\begin{aligned} K_n(z,a)&=v_n^*R_n^*(\bar{z})H_{1,n}^{-1}R_n(a)v_n \end{aligned}$$

(4.25)

$$\begin{aligned}&=\frac{P^*_{1,n}(\bar{z})\widehat{H}_{1,n}^{-1}P_{1,n+1}(a) -P^*_{1,n+1}(\bar{z})\widehat{H}_{1,n}^{-1}P_{1,n}(a)}{z-a}\end{aligned}$$

(4.26)

$$\begin{aligned}&=(M_0+\cdots +M_n)+\cdots +(-1)^{n}(z-a)^j M_0L_0M_1\cdots L_{n-1}M_n \end{aligned}$$

(4.27)

$$\begin{aligned}&=-\Gamma _{2,n}^*(\bar{z},a) \Theta _{2,n}^{*^{-1}}(a,a). \end{aligned}$$

(4.28)

Proof

Equality (4.25) was proven in [3, Proposition 4.8] as an additive representation of block matrix $\gamma ^{(2j)}$; see (1.12). Equality (4.26) is verified using Definition 11 and the proof of [28, Lemma 3]. Equality (4.27) readily follows from (4.7). Finally, equality (4.28) is a consequence of (1.17). $\square$

Equality (4.25) for $a=0$ first appeared in [24].

Observe that equalities (4.25)–(4.28) are also valid if instead of real a one uses a complex variable w. This generalization will be considered elsewhere.

Now we introduce new CD type kernels.

Definition 12

Let $[a,b]\subset \mathbb R$, and let $(s_j)_{j=0}^{2n}$, $(s_j)_{j=0}^{2n+1}$ be Hausdorff positive definite sequences. Let $Q_{2,j}$, $\Gamma _{1,j}$, $\Theta _{2,j}$, $\widehat{H}_{2,j}$, $\widehat{K}_{1,j}$ and $\widehat{K}_{2,j}$ as in Definition 10, 9, (1.10), (1.24) and (1.25), respectively. Define

$$\begin{aligned} K^{(1)}_{n-1}(z,a,b):=&\,s_0+\sum _{j=0}^{n-1} Q_{2,j}^*(\bar{z})\widehat{H}_{2,j}^{-1}Q_{2,j}(a),\\ K^{(2)}_n(z,a,b):=&\sum _{j=0}^n \Gamma _{1,j}^*(\bar{z})\widehat{K}_{1,j}^{-1}\Gamma _{1,j}(a),\\ K^{(3)}_n(z,a):=&\sum _{j=0}^n \Theta _{2,j}^*(\bar{z})\widehat{K}_{2,j}^{-1}\Theta _{2,j}(a). \end{aligned}$$

Proposition 10

Under the same conditions as Proposition 9 and Definition 12 the following equalities hold:

$$\begin{aligned} K^{(1)}_{n-1}(z,a,b)&=s_0+ (u_{2,n-1}^*-z s_0 v_{n-1}^*)R_{n-1}^*(\bar{z}) H_{2,n-1}^{-1}R_{n-1}(a)(u_{2,n-1}-av_{n-1}s_0) \nonumber \\&=\Theta _{1,n}^*(\bar{z},b) \Gamma _{1,n}^{*^{-1}}(a,b), \end{aligned}$$

(4.29)

$$\begin{aligned} K^{(2)}_{n}(z,a,b)=\,&v_n^*R_n^*(\bar{z})K_{1,n}^{-1}R_n(a)v_n =P_{2,n}^*(\bar{z},a,b) Q_{2,n}^{*^{-1}}(a,b,a), \end{aligned}$$

(4.30)

$$\begin{aligned} K^{(3)}_{n}(z,a)=\,&\widetilde{u}_{2,n}^*R_n^*(\bar{z})K_{2,n}^{-1}R_n(a) \widetilde{u}_{2,n} =-Q_{1,n+1}^{*}(\bar{z})P_{1,n+1}^{*^{-1}}(a). \end{aligned}$$

(4.31)

Proof

The first equality of (4.29) follows from (1.13) and [4, Equation (3.15)] The second equality of (4.29) follows from [4, Equation (3.19)] or (1.18).

Furthermore, the first equality of (4.30) [resp. (4.31)] follows from (1.27) [resp. (1.28)] and [34, p. 87]. Finally, the second equality of (4.30) [resp. (4.31)] follows from [4, Equality (3.32)] and [4, Equality (3.31)], respectively. $\square$

Further properties of the CD type kernels will be considered elsewhere.