Mixed periodic Jacobi continued fractions

Abstract

Let b0 be a positive real number andbe a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as followswhere An(z)/Bn(z) is the n-th Padé approximant of φ(z).

Decompositions of the free additive convolution

“…and it is understood that if ω m = 0 for some m, then the fraction terminates and, for convenience, we set ω n = α n = 0 for all n > m. In examples, we will mainly characterize measures by giving their Jacobi sequences and refer the reader to [13] for details and explicit measures. Let us also introduce the finite approximations of continued fractions which will be used in computations.…”

“…where P (z) = (z − 2α) 2 , A 2 = 2ω 2 , Λ 1 (z) = 3z − 2α and Γ 1 (z) = z 2 − zα + ω 2 , using the notation of [13,Theorem 3.4] (with N = 1, M = 1). Thus μ --μ is the measure with the absolutely continuous part given by the density…”

Spectral properties of certain tridiagonal matrices

“…The analytical study of infinite tridiagonal matrices (infinite Jacobi matrices, regarded as operators acting in ℓ 2 , the space of the square summable sequences of complex numbers) was considered before by several authors. For instance, in [8,9,11] in connection with the Theory of Toda Lattices, as well as in [14,18,19,23,25,26], where the spectrum of the corresponding Jacobi operators was studied. We also point out that a matrix theoretic approach to the problem concerning the study of the spectral properties of k−Toeplitz matrices has been presented in works by S. Serra Capizzano and D. Fasino [12,24].…”

A Survey on Spectra of infinite Graphs

“…Furthermore, the computation of the spectral measure n 0 0 (dX) also becomes an easier task. Compare with [71]; see also [73,74].…”