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Darboux transformations of Jacobi matrices and Padé approximation. (English) Zbl 1227.47018

The sequence \(\{P_j\}_{j=0}^\infty\) of monic orthogonal polynomials with respect to a probability measure \(\sigma\) on \({\mathbb R}\) (with infinite support and all moments finite) is known to satisfy the recurrence relation
\[ \lambda P_j(\lambda) = P_{j+1}(\lambda) + b_j P_j(\lambda) +c_{j-1} P_{j-1}(\lambda) \]
with initial conditions
\[ P_{-1}(\lambda) = 0,\;P_0(\lambda) = 1, \]
where \(b_j\in{\mathbb R}\) and \(c_j>0,\,j\in{\mathbb Z}_{+}\). The companions to these so-called polynomials of the first kind are the polynomials of the second kind \(Q_j\), satisfying the same recurrence relation, but with the initial conditions
\[ Q_{-1}(\lambda) = -1,\;Q_0(\lambda) = 0. \]
The semi-infinite tridiagonal matrix \(J\) satisfying
\[ J p(\lambda) = \lambda p(\lambda), \]
where \(p = (P_0,P_1,\dots,P_j,\dots)^{\text{T}}\), is said to be the monic Jacobi matrix associated with \(\sigma\).
Using the unique \(LU\)-factorization \(J=LU\) (which exists if and only if \(P_j(0)\not= 0,\,j\in{\mathbb Z}_{+}\)), the Christoffel transformation of \(J\) is defined by
\[ J = LU\mapsto J_C:= UL \]
(\(J_C\) is the tridiagonal matrix associated with \(td\sigma(t)\)). If \(s_{-1}\) is a real number with \(Q_j(0)-s_{-1}P_j(0)\not= 0,\,j\in{\mathbb Z}_{+}\), then \(J\) has the \(UL\)-factorization \(J=UL\) and the Geronimus transformation of \(J\) with parameter \(s_{-1}\) is
\[ J=UL\mapsto J_G:= LU. \]
These two transformations are also called the Darboux transformations.
In the paper under review, these transformations are extended beyond the conditions given: for an arbitrary monic Jacobi matrix, a pair of lower and upper triangular block matrices \(L\) and \(U\) is constructed such that \(J=LU\) and
\[ {\mathcal J}_C=UL \]
is a monic generalized Jacobi matrix, i.e., it is a tridiagonal block matrix with blocks as specified in their Definition 2.4.
Similarly, with a free parameter \(s_{-1}\), the \(UL\)-factorization \(J=\mathcal{ U L}\) with lower and upper triangular block matrices is found and the Geronimus transform is also a tridiagonal block matrix
\[ {\mathcal J}_G=\mathcal{ L U}. \]
The associated Nevanlinna functions are
\[ {\mathcal F}_C(\lambda)=\lambda\int_{{\mathbb R}}\,{d\sigma(t)\over t-\lambda}+1,\;{\mathcal F}_G(\lambda)=-{s_{-1}\over \lambda}+{1\over \lambda}\,\int_{{\mathbb R}}\,{d\sigma(t)\over t-\lambda}. \]
After this introduction, the paper consists of the following sections:
2.
Preliminaries. Several classes of Nevanlinna functions and normal indices of the sequence of coefficients of the asymptotic expansion, Schur transform, \(P\)-fraction, introduction of an indefinite inner product.
3.
The Christoffel transform and its inverse. Explicit forms of the relevant block structure for the matrices are given.
4.
The Geronimus transform and its inverse. Explicit forms of the relevant block structures, connection with normal indices.
5.
Generalized Cholesky decomposition. Explicit form of the block structure of the decomposition \(J=L\Lambda L^{\text{T}}\) using normal indices.
6.
Convergence of Padé approximants. Here, the probability measures are supposed to be supported on \([-1,1]\) and convergence results in terms of properties of the Darboux transform are given for sequences of diagonal approximants.
7.
Examples. Two examples are given, one exhibiting a bounded Jacobi matrix having an unbounded Christoffel transform.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
30E05 Moment problems and interpolation problems in the complex plane
41A21 Padé approximation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
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References:

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