On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

Abstract

We study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Pade approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Pade approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker Gammel Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the n th root asymptotic behavior of Pade denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of``spurious'' Pade poles in this set may be bounded. Academic Press

The linear pencil approach to rational interpolation

“…Subsequently, we present in Theorem 2.10 of §2.3 an alternate proof for a Favard-type theorem based on orthogonality properties of associated rational functions, which yields in Corollary 2.12 a simple proof for the fact that the convergents of our continued fractions are indeed multipoint Padé approximants of the m-function of our linear pencil. In §3 we generalize the above-mentioned results of [12,Theorem 3.10], [11,Theorem 3.1], and [11,Theorem 4.4], on the convergence of Padé approximants at infinity in terms of complex Jacobi matrices to the more general case of multi-point Padé approximants in terms of linear pencils zB − A. The aim of §4 is to explore LU and U L decompositions of our linear pencil, and the link to biorthogonal rational functions.…”

“…As explained already before, in general one may not expect convergence of m [0:n] to m locally uniformly in ρ(A, B) since there might be socalled spurious poles in ρ(A, B). One strategy of overcoming the problem of spurious poles is to allow for exceptional small sets, as done in [11,Theorem 3.1] for complex Jacobi matrices where convergence in capacity is established. We may generalize these findings for linear pencils, where again we follow the lines of the alternate proof presented in [9,Theorem 4.7].…”

“…There is no longer a natural candidate for the spectrum of A, but it is still possible to characterize the spectrum in terms of some asymptotic behavior of the Padé denominators q n (z) and the linearized error functions r n (z) = q n (z)φ(z) − p n (z) [3,12,10], see also [8,17,16] for more general banded matrices. Convergence outside the numerical range was established in [12], and convergence in capacity in the outer connected component of the resolvent set in [11]. We refer the reader to [9] for some recent summary on complex Jacobi matrices, including some open questions partially solved in [7].…”

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

“…then it is related to the matrix sequence of polynomials {V m } through the recurrence relation (9). From now on this block matrix is said to be the 2 × 2 block Jacobi matrix associated with the above matrix polynomial sequences.…”

Complex Jacobi matrices and quadrature rules

“…This statement is the famous Favard theorem which proof can be found in [4]. 1 For any given two sequences {α k } and {β k }, β k = 0, the following infinite matrix can be associated…”